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America n Mathematica l Societ y
Colloquiu m Publication s
Volum e 23
Orthogona l
Polynomial s
Gabo r Szeg o
America n Mathematica l Societ y
Providence , Rhod e Islan d
2000 Mathematics Subject
Classification.
Primar
y 33-XX .
Library o f Congres s Car d Numbe r 39-3349 7
ISBN 0-821 8-1 023- 5
ISSN 0065-925 8
C o p y i n g a n d r e p r i n t i n g . Individua l reader s o f thi s publication , an d nonprofi t librarie s
acting fo r them , ar e permitte d t o mak e fai r us e o f th e material , suc h a s t o cop y a chapte r fo r us e
in teachin g o r research . Permissio n i s grante d t o quot e brie f passage s fro m thi s publicatio n i n
reviews, provide d th e customar y acknowledgmen t o f th e sourc e i s given .
Republication, systemati c copying , o r multipl e reproductio n o f an y materia l i n this publicatio n
is permitte d onl y unde r licens e fro m th e America n Mathematica l Society . Request s fo r suc h
permission shoul d b e addresse d t o th e Acquisition s Department , America n Mathematica l Society ,
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© 1 93 9 b y th e America n Mathematica l Society . Al l right s reserved .
Reprinted wit h corrections , 200 3
Printed i n th e Unite d State s o f America .
@ Th e pape r use d i n thi s boo k i s acid-fre e an d fall s withi n th e guideline s
established t o ensur e permanenc e an d durability .
15 1 41
1 31 2
4 131211 1 0 09
TO MY WIF E
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PREFACE
Recent year s hav e see n a grea t dea l o f progres s i n th e field o f orthogona l
polynomials, a subjec t closel y relate d t o man y importan t branche s o f analysis .
Orthogonal polynomial s ar e connecte d wit h trigonometric , hypergeometric ,
Bessel, an d ellipti c functions , ar e relate d t o th e theor y o f continue d fraction s
and t o importan t problem s o f interpolatio n an d mechanica l quadrature , an d
are of occasiona l occurrenc e in the theorie s of differentia l an d integra l equations .
In addition , the y furnis h comparativel y genera l an d instructiv e illustration s o f
certain situation s in the theor y o f orthogona l systems . Recently , som e of thes e
polynomials hav e bee n show n t o b e o f significanc e i n quantu m mechanic s an d
in mathematica l statistics .
The origin s o f th e subjec t ar e t o b e foun d i n th e investigatio n o f a certai n
type o f continue d fractions , bearin g the name o f Stieltjes. Specia l cases of thes e
fractions wer e studied b y Gauss , Jacobi, Christoffel , an d Mehler , amon g others ,
while mor e genera l aspect s o f thei r theor y wer e give n b y Tchebichef , Heine ,
Stieltjes, an d A . Markoff .
Despite th e clos e relationshi p betwee n continue d fraction s an d th e proble m
of moments , an d notwithstandin g recen t importan t advance s i n thi s latte r
subject, continue d fraction s hav e bee n graduall y abandone d a s a startin g poin t
for th e theor y o f orthogona l polynomials . I n thei r place , th e orthogona l
property itself ha s been taken as basic, and it is this poin t of view which has been
adopted i n th e followin g expositio n o f th e subject . Choosin g thi s sam e basi c
property, w e discus s certai n specia l orthogona l polynomials , whic h hav e bee n
treated i n grea t detai l independentl y o f th e genera l theory , an d indeed , eve n
before thi s theor y existe d a t all . I n thi s connectio n w e ad d th e name s o f La place, Legendre , Fourier , Abel , Laguerre , an d Hermit e t o thos e previousl y
mentioned.
As regard s treatise s o n th e subject , w e not e tha t th e onl y systemati c treat ment thu s fa r give n i s foun d i n J . Shohat' s monograph , Theorie Ge'ne'rale des
Polynomes Orthogonaux de Tchebichef, Memorial de s Sciences Math6matiques ,
Paris, 1 934 . Limitation s o f spac e hav e compelle d tha t wor k t o b e brief , an d
consequently, i t doe s no t ente r int o a detaile d treatmen t o f man y problem s
which hav e bee n especiall y advance d i n recen t years . I t ha s therefor e seeme d
desirable t o attemp t a ne w an d detaile d developmen t o f th e mai n idea s o f thi s
field, devoting , i n particular , som e spac e to recent investigations of th e distribu tion o f th e zeros , o f asymptoti c representations , o f expansio n problems , an d o f
certain question s o f interpolation an d mechanica l quadrature .
In what follows, we are concerned partly with the general theory of orthogona l
polynomials, an d partl y wit h th e stud y o f specia l classe s o f thes e polynomials .
As might b e expected , w e have mor e exhaustiv e result s for thes e specia l classes ,
and w e cite a s a n instanc e th e classica l polynomial s satisfyin g linea r differentia l
v
PREFACE
VI
equations o f th e secon d order . Also , whe n th e primar y importanc e o f thes e
special classe s i n application s i s taken int o account , i t shoul d no t b e a t al l sur prising tha t th e presen t boo k i s mainl y devote d t o thei r study . Th e genera l
theory, however , a s develope d i n Chapter s X I I an d XIII , doubtles s represent s
the most importan t progres s made i n recent years .
In th e presen t work , n o claim is made for completenes s o f treatment. O n th e
contrary, th e ai m ha s purposel y bee n t o mak e th e materia l suggestiv e rathe r
than exhaustive . A n attemp t ha s bee n mad e t o indicat e th e mai n an d charac teristic method s an d t o poin t ou t th e relatio n o f thes e t o som e genera l idea s i n
modern analysis . A s a rule , preferenc e ha s bee n give n t o thos e topic s t o whic h
we wer e abl e t o mak e som e new , thoug h modest , contributions , o r whic h w e
could present i n a new setting. Thu s th e book contain s a number o f results no t
previously published, som e of which originated severa l years ago. Fo r instance ,
we have include d a discussio n o f th e Cesar o summabilit y o f th e Jacob i serie s a t
the end-point s o f the orthogonalit y interva l (th e method use d here i s of interes t
even i n th e classica l cas e o f Legendr e series) . Further , a ne w an d simple r ap proach ha s bee n give n t o S . Bernstein' s asymptoti c formul a fo r orthogona l
polynomials. W e als o refe r t o certai n detail s o f mino r importance , suc h as :
simplifications an d addition s i n th e asymptoti c investigatio n o f Jacob i an d
Laguerre polynomial s an d i n th e discussio n o f th e expansion s i n term s o f thes e
polynomials; the discussio n o f the case s in which the Jacobi differentia l equatio n
has onl y polynomia l solutions ; the evaluatio n o f th e numbe r o f zero s of genera l
Jacobi polynomials in the intervals[ — <» , — 1],[— 1 , + 1 ] , [+ 1 , + oo];ane w
proof o f the Heine-Stieltjes theore m o n linear differential equation s of the secon d
order wit h polynomia l coefficient s an d polynomia l solutions , an d s o on .
In general , w e hav e preferre d t o discus s problem s whic h ma y b e state d an d
treated simply , an d whic h coul d b e presente d i n a mor e o r les s complet e form .
This wa s th e mai n reaso n fo r devotin g n o spac e t o th e extremel y interestin g
arithmetic an d algebrai c propertie s o f orthogona l polynomials , suc h as , fo r
instance, th e recen t importan t investigation s o f I . Schu r concernin g th e irre ducibility an d relate d propertie s o f Laguerr e an d Hermit e polynomials . Fur thermore, we have attached grea t importance to the idea of replacing incomplet e
and overlappin g theorems , scattere d i n th e literature , b y complet e result s
involving only intrinsic or necessary restrictions. W e have also tried t o exploit ,
as fa r a s seeme d t o b e a t al l possible , definit e methods , suc h as , fo r instance ,
Sturm's methods i n differentia l equation s (se e §§6.3 , 6.31 , 6.32, 6.83) .
A complete treatment o f Legendre polynomials was not feasible, an d probabl y
not desirable , i n th e framewor k o f th e genera l theory . Besides , ther e ar e al ready complet e treatise s o n spherica l an d othe r harmonics. 1 W e hav e selecte d
and considere d onl y thos e propertie s o f Legendr e polynomial s whic h ar e th e
starting point s o f generalization s t o ultraspherical , Jacobi , o r t o more genera l
polynomials. Anothe r subjec t whic h coul d no t b e include d wa s Stieltjes '
1
For instance, E . W . Hobson 1 (se e bibliography) .
PREFACE
vii
problem o f moments , whic h ha s bee n omitte d i n spit e o f it s grea t interest ;
for thi s subjec t woul d hav e necessitate d th e developmen t o f a complicate d
apparatus o f result s an d methods . Orthogona l polynomial s o f mor e tha n on e
variable als o have no t bee n treated. 2
The boo k i s base d o n a cours e give n a t Washingto n Universit y durin g th e
academic yea r 1 935-1 936 . Acquaintanc e wit h th e genera l idea s an d method s
of th e theor y o f function s o f rea l an d comple x variable s i s naturall y required .
Occasionally, Stieltjes-Lebesgu e an d Lebesgu e integrals ar e considered . I n th e
greater part o f the book, however, these integrals have been avoided , and , excep t
in a ver y fe w places , n o detaile d propertie s o f the m wer e used .
The problem s a t th e en d o f thi s boo k are , wit h fe w exceptions , no t new , an d
they are not interconnected a s are, for instance, those in Polya-Szego's Aufgaben
und Lehrsdtze. The y ar e mor e o r les s supplementar y i n characte r an d serv e a s
illustrations an d exercises ; they sometime s diffe r widel y fro m on e anothe r bot h
as to subject an d method .
The list o f reference s i s not complete ; it contain s onl y original memoirs, a fe w
text books of primary importance, and monographs to which references ar e mad e
in the text .
For th e suggestio n o f preparin g a boo k o n orthogona l polynomial s fo r th e
Colloquium Publications , I a m indebte d t o Professo r J . D . Tamarkin , wh o ha s
also participate d i n th e presen t wor k b y offerin g a grea t numbe r o f valuabl e
suggestions. I t i s wit h th e greates t gratitud e tha t I mentio n hi s friendl y
interest.
1 hav e als o receive d valuabl e advic e fro m m y friend s an d teacher s L . Feje r
(Budapest), an d G . P61 y a (Zurich) . M y colleague s P . Erdo s (Manchester) ,
G. Grtinwal d (Budapest) , W . H . Roeve r (St . Louis) , A . Ros s (St . Louis) , J .
Shohat (Philadelphia) , an d P . Tura n (Budapest ) gav e generousl y an d unstint ingly o f thei r time . F . A . Butter, Jr . (a t presen t i n Lo s Angeles) collaborate d
with me i n th e preparatio n o f th e manuscript . Thi s last ai d wa s made possibl e
through a gran t fro m th e Rockefelle r Researc h Fun d o f Washingto n Universit y
(1936-1937). M y studen t L . H . Kante r als o rendere d valuabl e assistanc e i n
the preparatio n o f th e manuscript .
My gratitud e fo r th e encouragemen t an d hel p o f thes e friends, colleagues , and
institutions ca n hardl y b e measure d b y an y forma l acknowledgment . Lastly ,
I wis h t o expres s t o th e America n Mathematica l Societ y m y grea t appreciatio n
for th e inclusio n o f th e presen t boo k i n it s Colloquiu m Series .
G. SZEG O
WASHINGTON UNIVERSITY ,1 938 .
2
Cf. th e bibliograpn y i n Jackso n 8 , p . 423.
PREFACE T O T H E REVISE D EDITIO N
The first printin g of this book publishe d i n 1 93 9 wa s abou t exhauste d i n 1 948 .
Reprinting wa s arranged the n bu t fo r variou s reason s n o chang e i n th e tex t wa s
made. Durin g th e pas t twent y year s sinc e th e preparatio n o f th e origina l
edition wa s completed , considerabl e proges s wa s mad e i n thi s field . A glanc e
at th e pertinen t sectio n o f th e Mathematica l Review s suggest s tha t th e interes t
in thi s topi c i s stil l ver y muc h alive . Systemati c treatmen t o f orthogona l
polynomials ha s bee n incorporate d i n variou s moder n text s publishe d i n th e
meantime. We refe r onl y t o th e Higher Transcendental Functions publishe d b y
the Batema n Manuscrip t Projec t Staf f (cf . i n particular , vol . 2 , Chapte r X ,
edited b y Professo r A . Erdelyi) , an d t o th e boo k o f F . Tricomi , Vorlesungen
uber Orthogonalreihen (Chapter s IV—VI) .
Recently th e counci l o f th e America n Mathematica l Societ y ha s authorize d
the autho r t o prepar e a revise d editio n o f th e book , addin g a moderat e amoun t
of material i n orde r t o brin g i t u p t o date . Naturally , limitation s o f spac e an d
time did no t allo w includin g al l ne w result s (or , fo r tha t matter , th e ol d one s
which were missing fro m th e origina l edition) . Onl y a fe w particularl y interest ing new item s hav e bee n adde d a s well a s som e detail s whic h deserv e attentio n
because o f eleganc e o f th e metho d o r originalit y o f ideas . W e mentio n her e i n
particular th e important Pollacze k polynomials ; they ar e treated i n an Appendix .
Further ne w materia l wa s incorporate d i n th e for m o f Problem s an d Exercises .
New bibliographi c item s hav e bee n included , agai n i n a rathe r selectiv e way .
Finally, misprint s hav e bee n correcte d an d numerou s mino r improvement s an d
additions made .
The author recollect s again, as was stated i n the Preface o f 1938, that th e preparation of this book was suggested t o hi m b y th e lat e Professo r J . D . Tamarkin .
Since his untimely deat h i n 1 94 5 his name is not to o frequentl y mentioned . I t i s
justified an d probabl y necessar y t o remin d th e younge r mathematica l gener ation, i n th e rus h o f moder n developments , ho w muc h America n mathematic s
owes t o hi s grea t energ y an d far-sighte d intelligence .
STANFORD UNIVERSITY ,1 95 8
G. SZEG G
PREFACE T O TH E THIR D EDITIO N
The interes t o f th e mathematica l communit y fo r orthogona l polynomials ,
classical an d non-classical , i s stil l no t entirel y exhausted . Durin g th e pas t
years I lecture d abou t thi s subjec t severa l time s a t Stanford . Th e attendant s
of th e cours e wer e uppe r divisio n an d graduat e students , specializin g i n
mathematics, mathematica l statistics , calculu s o f probability , etc .
Only mino r change s hav e bee n mad e i n th e text . I ow e numerou s im provements an d correction s t o variou s friend s an d colleagues . I
mentio n
particularly Professo r Pau l Tura n (Budapest , Hungary ) an d Professo r
Lee Lorc h (Edmonton , Canada) . Ne w references , publishe d i n th e tim e
interval 1 958-1 966 , hav e bee n included .
STANFORD UNIVERSITY , 1 96 6 G
. SZEG O
PREFACE T O TH E FOURT H EDITIO N
Again th e America n Mathematica l Societ y ha s take n th e initiativ e t o
reprint th e presen t book , allowin g som e mino r change s an d ne w material .
Among th e person s intereste d i n th e fiel d o f orthogona l polynomial s wh o
have contribute d t o thes e change s an d additions , I mentio n wit h particula r
indebtedness m y frien d an d colleagu e Professo r Richar d Aske y (Madison ,
Wisconsin) an d th e ver y activ e an d origina l grou p o f mathematician s aroun d
him. A ver y importan t se t o f lecture s b y Aske y entitled , "Orthogona l
Polynomials an d Specia l Functions/ ' reache d m e to o lat e t o b e incorporate d
in th e presen t edition .
Further materia l ha s bee n furnishe d b y Professo r Pau l Tura n (Rudapest ,
Hungary) an d Professo r Le e Lorc h (Toronto , Canada) . Ne w problem s an d
exercises hav e als o bee n included . Pete r Szeg o (Redwoo d City , California )
gave m e valuabl e assistanc e i n preparin g th e presen t manuscript .
My gratitud e goe s t o al l thes e friend s an d colleagues .
STANFORD UNIVERSITY , 1 97 5 G
. SZEG O
XI
TABLE O F CONTENTS
PAGE
PREFACE v
PREFACE TO THE REVISED EDITION i
x
PREFACE TO THE THIRD EDITIO N x
i
PREFACE T O THE FOURT H EDITIO N x
i
CHAPTER I . PRELIMINARIE
S
CHAPTER II . DEFINITIO
N O F ORTHOGONA L POLYNOMIALS ; PRINCIPA L EXAMPLE S
CHAPTER III . GENERA
3
L PROPERTIE S O F ORTHOGONA L POLYNOMIAL S 3
CHAPTER IV . JACOB
8
I POLYNOMIAL S 5
CHAPTER V . LAGUERR
CHAPTER VI . ZERO
. . .2
8
1
E AN D HERMIT E POLYNOMIAL S 0
0
S O F ORTHOGONA L POLYNOMIAL S I l
CHAPTER VII . INEQUALITIE
CHAPTER VIII . ASYMPTOTI
9
C PROPERTIE S O F TH E CLASSICA L1
POLYNOMIAL S 9 1
CHAPTER I X . EXPANSIO
N PROBLEM S ASSOCIATE D WIT H TH E CLASSICA L POLYNOMIAL S .
CHAPTER X . REPRESENTATIO
24 4
N O F POSITIV E FUNCTION S .
CHAPTER X I . POLYNOMIAL
27 4
S ORTHOGONA L O N TH E UNI T CIRCL E 28
CHAPTER X I I . ASYMPTOTI
CHAPTER X I I I . EXPANSIO
l
S 5
7
C PROPERTIE S O F GENERA L ORTHOGONA L POLYNOMIAL S .
N PROBLEM
S ASSOCIATE
D WIT
H GENERA
.
.
L ORTHOGONA
29 6
L
POLYNOMIALS 3
CHAPTER XIV . INTERPOLATIO
CHAPTER XV . MECHANICA
CHAPTER XVI . POLYNOMIAL
3
N 32
9
L QUADRATUR E 34
9
S ORTHOGONA L O N A N ARBITRAR Y CURV E 36
4
PROBLEMS AN D EXERCISE S 37
7
FURTHER PROBLEM S AN D EXERCISE S 38
7
APPENDIX 39
3
LIST O F REFERENCE S 40 1
FURTHER REFERENCE S 4
6
INDEX .
.
xiii
42 5
LIST O F REFERENCES 73
ABEL, N . H
.
*1. Oeuvres Completes. Vol.2
, 1 881 .
ACHIESER, N . I .
1. fiber eine Eigenschaft der "elliptischen" Polynome.
Communication
Mathematique d e Kharkoff , (4) , vol . 9 (1 934) , pp . 3-8 .
2. Verallgemeinerung einer
Korkine-Zolotareffschen Minimum-Aufgabe.
vol. 1 3 (1 936) , pp . 3-1 4 .
s d e l a Societ y
Ibid.
, (4) ,
ADAMOFF, A .
1. On the asymptotic expansion of the polynomials U n(x) = = e ax l2 dn[e~ax f2 ]/dxn for
large values of n (i n Russian) . Annal s o f th e Polytechni c Institut e o f St . Peters burg, vol . 5 (1 906) , pp . 1 27-1 43 .
2. Expansions of an Arbitrary Function of a Single Real Variable in Series of Functions
of a Preassigned Kind (i n Russian) . Thesis . St . Petersburg , 1 907 , 1 9 1 pp .
BANACH, S .
*1. Theorie des Operations Lineaires. Warszawa-Lw6w
, 1 932 .
BERNSTEIN, S .
*1. Legons sur les Proprietes Extremales et la Meilleure Approximation des
Fonctions
Analytiques d'une Variable R'eelle. Paris , 1 926 .
2. Sur les polynomes orthogonaux relatifs a un segment fini. Journa l d e Math6matiques ,
(9), vol . 9 (1 930) , pp . 1 27-1 77 ; vol . 1 0 (1 931 ) , pp . 21 9-286 .
3. Sur une classe de polynomes orthogonaux. Communication s d e l a Societ e Mathe matique d e Kharkoff , (4) , vol . 4 (1 930) , pp . 79-93 . Complement. Ibid. , vol . 5
(1932), pp . 59-60 .
4. Sur la limitation des valeurs d'un polynome P n(x) de degre n sur tout un segment par
ses valeurs en (n + 1 ) points du segment. Bulleti n d e l'Academi e de s Science s
de TURSS , 1 931 , pp . 1 025-1 050 .
BLUMENTHAL, O .
1. Ueber die Entwicklung einer willkiirlichen Funktion nach den Nennern des Kettenbruches fur J^_ [#(£)/(z — £)]d£. Inaugural-Dissertation . Gottingen , 1 898 .
BOCHNER, S .
1. fiber Sturm-Liouvillesche Polynomsysteme.
(1929), pp . 730-736 .
Mathematisch
e Zeitschrift , vol . 2 9
BOTTEMA, O .
1. Die Nullstellen der Hermiteschen Polynome. Koninklijk e Akademi e va n Weten schappen t e Amsterdam , Proceedings , vol . 3 3 (1 930) , pp . 495-503 .
2. Die Nullstellen gewisser durch Rekursionsformeln definierlen
Polynome. Ibid.
,
vol. 3 4 (1 931 ) , pp . 681-691 .
BRAUER, A .
1. fiber die Nullstellen der
(1932), pp . 87-89 .
Hermiteschen Polynome.
Mathematisch
e Annalen , vol . 1 0 7
BRUNS, H .
1. Zur Theorie der Kugelfunclionen. Journa
matik, vol . 9 0 (1 881 ) , pp . 322-328 .
l fli r di e rein e un d angewandt e Mathe -
B'UELL, C . E .
1. The zeros of Jacobi and related polynomials. Duk
(1936), pp . 304-31 6 .
73
e Mathematica l Journal , vol . 2
The asterisk s indicat e item s no t dealin g wit h orthogona l polynomials .
401
402
LIST O F R E F E R E N C E S
CARATHEODORY, C.
*1. Conformal Representation. Cambridge
, 1 932 .
CARLEMAN, T .
1. Vber die Approximation analytiscker Funktionen durch lineare Aggregate von vorgegebenen Potenzen. Arki v fo r Matematik , Astronom i oc h Fysik , vol . 1 7 (1 923) ,
no. 9 , 3 0 pp .
CHRISTOFFEL, E . B
.
1. Vber die Gaussische Quadratur und eine Verallgemeinerung derselben. Journa
die rein e un d angewandt e Mathematik , vol . 5 5 (1 858) , pp . 61 -82 .
l fii r
CRAMER, H .
1. On some classes of series used in mathematical statistics. Compte
s Rendu s d u Sixi eme Congre s de s Mathematieien s Scandinaves , Stockholm , 1 926 , pp . 399-425 .
DARBOUX, G .
1. Memoire sur Vapproximation des
fonclions de trhs grands nombres. Journa
Mathematiques, (3) , vol . 4 (1 878) , pp . 5-56 , 377-41 6 .
DIRICHLET, G . L
ld e
.
1. Sur les series dont le terme general depend de deux angles, el qui servent a exprimer des
fonclions arbitraires entre des limites donnees. Journa l fii r di e rein e un d ange wandte Mathematik , vol . 1 7 (1 837) , pp . 35-56 .
DOETSCH, G .
1. Integraleigenschaften der
Hermiteschen Polynome.
Mathematisch e Zeitschrift ,
vol. 3 2 (1 930) , pp . 587-599 .
2. Die in der Statistik seltener Ereignisse auftreienden Charlierschen Polynome und
eine damit zusammenhdngende Differentialdifferenzengleichung. Mathematisch
e
Annalen, vol . 1 0 9 (1 933) , pp . 257-266 .
D u BOIS-REYMOND , P .
*1. Untersuchungen uber die Convergenz und Divergenz der Fourierschen Darstellungsformeln. Abhandlunge n de r Akademi e Munchen , vol . 1 2 (1 876) , pp . 1 -1 03 .
ERD^LYI, A .
1. Vber eine Integraldarstellung der Mk.m-Funktionen und ihre asymptotische Darstellung fur grosse Werte von 9? C k. Mathematisch e Annalen , vol . 1 1 3 (1 936) , pp .
357-362.
2. Vber eine erzeugende Funktion von Produkten Hermilescher Polynome. Mathe matische Zeitschrift , vol . 4 4 (1 938) , pp . 201 -21 1 .
E R D O S , P. , an d FELDHEIM , E .
1. Sur le mode de convergence pour Vinterpolation de Lagrange. Compte
PAcademie de s Sciences , Paris , vol . 20 3 (1 936) , pp . 91 3-91 5 .
s Rendu s d e
E R D O S , P. , an d T U R A N , P .
1. On interpolation. I
. Quadrature- and mean-convergence in the Lagrange interpolation. Annal s o f Mathematics , (2) , vol . 3 8 (1 937) , pp . 1 42-1 55 .
2. On interpolation. II
. On the distribution of the fundamental points of Lagrange
and Hermite interpolation. Ibid.
, vol . 3 9 (1 938) , pp . 703-724 .
EULER, L .
*1. Institutiones Calculi Integralis. Vol
Omnia. Ser . 1 , vol. 1 2 , p. 224.
. 2 , sect . I , chap . X , proble m 1 30. Opera
FABER, G .
1. Vber polynomische Entvncklungen.
Mathematisch e Annalen , vol . 5 7 (1 903) ,
pp. 389-408 .
*2. Vber die interpolatorische Darstellung
stetiger Funktionen. Jahresberich t de r
Deutschen Mathematiker-Vereinigung , vol . 2 3 (1 91 4) , pp . 1 92-21 0 .
3. Tschebyscheffsche Polynome. Journa l fii r di e rein e un d angewandt e Mathematik ,
vol. 1 5 0 (1 91 9) , pp . 79-1 06 .
4. Uber nach Polynomen forlschreilende Reihen. Sitzungsbericht
Akademie de r Wissenschaften , 1 922 , p p . 1 57-1 78 .
403
e de r Bayrische n
FATOU, P .
*1. Series trigonometriques et
p p . 335-400 .
series de Taylor. Act
a Mathematica , vol . 3 0 (1 906) ,
FAVARD, J .
1. Sur les polynomes de Tchebicheff. Compte
Paris, vol . 20 0 (1 935) , p p . 2052-2053 .
FEJ£R, L
s Rendu s d e TAcadeim e de s Sciences ,
.
*1. Sur les fonctions bornees et integrables. Compte s Rendu s d e FAcad6mi e de s Sci ences, Paris , vol . 1 3 1 (1 900) , p p . 984-987 .
*2. Unlersuchungen uber Fouriersche Reihen. Mathematisch e Annalen , vol . 5 8 (1 904) ,
p p . 51 -69 .
3. Asymptolikus brl'ekek
meghatdrozdsdrol. Mathematika i 6 s Term6szettudom£ny i
Ertesito, vol . 2 7 (1 909) , pp . 1 -33 .
4. Uber die Laplacesche Reihe. Mathematisch e Annalen , vol . 6 7 (1 909) , p p . 76-1 09 .
*5. Uber trigonomeirische Polynome. Journa l fu r di e rein e un d angewandt e M a t h e matik, vol . 1 4 6 (1 91 5) , p p . 53-82 .
6. Uber Interpolation. Nachrichte n de r Gesellschaf t de r Wissenschafte n z u Gottingen ,
1916, p p. 66-91 .
7. Uber die Lage der Nullstellen von Polynomen, die aus Minimumforderungen gewisser
Art entspringen. Mathematisch
e Annalen , vol . 8 5 (1 922) , p p . 41 -48 .
8. Uber die Summabilitdt der Laplaceschen Reihe durch arithmetische Mittel. Mathe matische Zeitschrift , vol . 2 4 (1 925) , p p . 267-284 .
9. Abschdtzungen fur die Legendreschen und verwandte Polynome. Mathematisch e
Zeitschrift, vol . 2 4 (1 925) , p p . 285-298 .
10. Uber Weierstrass'sche Approximation, besonders
durch Hermitesche Interpolation.
Mathematische Annalen , vol . 1 0 2 (1 930) , p p . 707-725 .
11. Die Abschdtzung eines Polynoms in einem Intervalle, wenn Schranken fur seine Werte
und ersten Ableitungswerte in einzelnen Punkten des Intervalles gegeben sind, und
ihre Anwendung auf die Konvergenzfrage Hermitescher Inlerpolationsreihen.
Mathematische Zeitschrift , vol . 3 2 (1 930) , p p. 426-457 .
12. Ullrasphdrikus polynomok
osszegerol. Matematika i e s Fizika i Lapok , vol . 3 8
(1931), p p . 1 61 -1 64 .
13. Lagrangesche Interpolation und
die zugehorigen konjugierten Punkte. Mathe matische Annalen , vol . 1 0 6 (1 932) , p p . 1 -55 .
14. Besiimmung derjenigen Abszissen eines Intervalles, fur welche die Quadratsumme der
Grundfunktionen der Lagrangeschen Interpolation im Intervalle ein moglichst
kleines Maximum besitzt. Annal i dell a Scuol a Normal e Superior e d i Pisa , (2) ,
vol. 1 (1 932) , p p . 3-1 6 .
15. Mechanische Quadraturen mit positiven Cotesschen Zahlen. Mathematisch e Zeit schrift, vol . 3 7 (1 933) , p p . 287-309 .
16. On the characterization of some remarkable systems of points of interpolation by means
of conjugate points. America n Mathematica l Monthly , vol . 4 1 (1 934) , p p . 1 -1 4 .
17. Potenzreihen mit mehrfach monotoner Koeffizientenfolge und ihre Legendre Polynome.
Proceedings o f t h e Cambridg e Philosophica l Society , vol . 3 1 (1 935) , p p . 307-31 6 .
18. A hatvdnysorrol es a vele kapcsolatos Legendre-fele tobbtagu'akrdL Mathematika i
es Term6szettudom£ny i Ertesito , vol . 5 3 (1 935) , p p . 1 -1 7 .
19. Besiimmung von Grenzen fur die Nullstellen des Legendreschen Polynoms aus der
Stieltjesschen Integraldarstellung desselben.
Monatsheft e fu r Mathemati k un d
Physik, vol . 4 3 (1 936) , p p . 1 93-209 .
20. Trigonomeirische Reihen und Potenzreihen mit mehrfach monotoner Koeffizientenfolge.
Transactions o f t h e America n Mathematica l Society , vol . 3 9 (1 936) , p p . 1 8-59 .
404
LIST O F REFERENCE S
FEJER, L. , an d SZEGO , G .
*1. Vber die monotone Konvergenz von Potenzreihen mit mehrfach monotoner Koeffizientenfolge. Prac e Matematyczno-Fizyczne , vol . 44 (1 935) , pp . 15-25.
FEKBTE, M .
*1. Vber die Verteilung der Wurzeln bei gewissen algebraischen Gleichungen mit ganzzahligen Koeffizienten. Mathematisch e Zeitschrift , vol . 1 7 (1923), pp . 228-249.
FELDHEIM, E .
1. Sur V orthogonalite des fonctions fondamentales de Vinterpolation de Lagrange.
Comptes Rendu s de PAcad6mie de s Sciences, Paris , vol . 203 (1936), pp. 650-652.
2. Sur le mode de convergence dans Vinterpolation de Lagrange. Compte s Rendu s de
l'Academie de s Sciences de l'URSS, vol. 14 (1937), pp. 327-331.
3. Sur Vorthogonalite des fonctions fondamentales, et sur la forte convergence en moyenne
des polynomes d'interpolation de Lagrange dans le cas des abscisses de Tchebychef.
Bulletin d e la Soci6t6 Mathematique d e France, vol. 65 (1937), pp. 1-40.
4. Theorie de la convergence des proced'es d 1 interpolation et de quadrature mecanique.
Memorial de s Sciences Mathematiques , vol . 95, Paris, 1939.
FRIEDRICHS, K .
1. On certain inequalities and characteristic value problems for analytic functions and for
functions of two variables. Transaction s of the American Mathematical Society ,
vol. 41 (1937), pp. 321-364.
FlJJIWARA, M .
1. On the zeros ofJacobi polynomials. Japanes e Journal of Mathematics, vol. 2 (1925),
pp. 1-2.
GALBRUN, H .
1. Sur un developpement d' une fonction a variable reelle en series de polynomes. Bul letin de la SociSte Mathematique d e France, vol . 41 (1913)/ pp. 24-47.
GAUSS, C . F .
*1. Summatio quarumdam serierum singularium. Werke. Vol . 2, pp. 9^45 .
2. Methodus nova integralium valores per approximationem inveniendi. Werke.
3, pp. 163-196.
Vol .
GEGENBAUER, L.
1. Vber einige bestimmte Integrate. Sitzungsbericht e de r mathematisch-natur wissenschaftlichen Klass e de r Akademie de r Wissenschaften i n Wien , Abteilun g
Ha, vol . 70 (1874), pp. 433-443.
2. Vber die Functionen C vn(x). Ibid. , vol . 75 (1877), pp. 891 -905 .
3. Vber die Functionen C vn{x). Ibid. , vol . 97 (1888), pp. 259-270.
4. Zur Theorie der hypergeometrischen Reihe. Ibid. , vol . 100 (1891), pp. 225-244 .
5. Das Additionstheorem der Functionen Cvn(x). Ibid. , vol. 102 (1893), pp. 942-950.
6. Zur Theorie der Functionen C vn{x). Denkschrifte n de r Akademi e de r Wissen schaften i n Wien, Mathematisch-naturwissenschaftlich e Klasse , vol . 48 (1 884) ,
pp. 293-316.
7. Einige Satze uber die Functionen Cvn(x). Ibid. , vol. 57 (1890), pp. 425-480.
G(U)ERONIMUS, J .
1. Sur le polynome multiplement monotone qui s'bcarte le moins de zero, dont un coefficient est donne. Bulleti n d e PAcad6mi e de s Science s d e TURSS , 1 929 , pp.
377-389.
2. Sur Vecart minimal quadratique de z'ero d fun polynome. Rendicont i de l Circol o
Matematico d i Palermo, vol. 54 (1930), pp. 298-313.
3. On some problems of Tchebychef. America n Journal of Mathematics, vol. 53 (1931),
pp. 597-604.
4. On a problem of M. J. Shohat. Ibid. , vol . 54 (1932), pp. 85-91.
5. On some extremal properties of polynomials. Annal s o f Mathematics, (2) , vol. 3 7
(1936), pp. 483-517.
405
GOTTLIEB, M . J .
1. Concerning some polynomials orthogonal on a finite or enumerable set of points.
American Journa l o f Mathematics , vol . 6 0 (1 938) , pp . 453-458 .
GRONWALL, T . H
.
1. Uber die Laplacesche Reihe. Mathematisch e Annalen , vol . 7 4 (1 91 3) , pp . 21 3-270 .
2. Uber die Summierbarkeit der Reihen von Laplace und Legendre. Ibid. , vol. 7 5 (1 91 4) ,
pp. 321 -375 .
GRUNWALD, G .
1. Uber Diver genzerscheinungen der Lagrangeschen Interpolationspolynome stetiger
Funktionen. Annal s o f Mathematics , (2) , vol . 3 7 (1 936) , pp . 908-91 8 .
GRUNWALD, G. , an d T U R A N , P .
1. Uber Interpolation. Annal
(1938), pp . 1 37-1 46 .
i dell a Scuol a Normal e Superior e d e Pisa , (2) , vol . 7
HAAR, A .
*1. Zur Theorie der orthogonalen Funktionensysteme. I
vol. 6 9 (1 91 0) , pp . 331 -371 .
2. Reihenentwicklungen nach Legendreschen Polynomen.
121-136.
. Mathematisch e Annalen ,
Ibid.
, vol . 7 8 (1 91 7) , pp .
HAHN, W .
1. Die Nullstellen der Laguerreschen und Hermiteschen Polynome. Inauguraldisserta
tion, Berlin . Schrifte n de s Mathematische n Seminar s un d de s Institut s fu r
angewandte Mathemati k de r Universita t Berlin , vol . 1 (1 933) , pp . 21 3-244 .
2. Bericht uber die Nullstellen der Laguerreschen und der Hermiteschen Polynome.
Jahresbericht de r Deutsche n Mathematiker-Vereinigung , vol . 4 4 (1 934) , pp . 21 5 236. Nachtrag. Ibid. , vol . 4 5 (1 935), ' p. 21 1 .
3. Uber die Jacobischen Polynome und zwei verwandte Polynomklassen. Mathe
matische Zeitschrift , vol . 3 9 (1 935) , pp . 634-638 .
4. Uber hbhere Ableitungen von Orthogonalpolynomen. Ibid
. vol . 4 3 (1 937) , p . 1 01 .
HAMBURGER, H .
1. Beitrdge zur Konvergenztheorie der Stieltjesschen Kettenbruche. Mathematisch e
Zeitschrift, vol . 4 (1 91 9) , pp . 1 86-222 .
2. Uber eine Erweiterung des Stieltjesschen Momentenproblems. Mathematisch
e An nalen, vol . 8 1 (1 920) , pp . 235-31 9 ; vol . 8 2 (1 920) , pp . 1 20-1 64 , 1 68-1 87 .
HARDY, G . H
.
1. Summation of a series of polynomials of Laguerre. Journa
matical Society , vol . 7 (1 932) , pp . 1 38-1 39 , 1 92 .
l o f th e Londo n Mathe -
HARDY, G . H. , LITTLEWOOD , J . E. , an d P 6 L Y A , G .
*1. Inequalities. Cambridge
, 1 934 .
HAUSDORFF, F .
*1. Momentprobleme fur ein endliches Intervall. Mathematisch
(1923), pp . 220-248 .
e Zeitschrift , vol . 1 6
HEINE, E .
1. Mittheilunguber Kettenbruche. Journa l fu r di e rein e un d angewandt e Mathematik ,
vol. 6 7 (1 867) , pp . 31 5-326 .
2. Die Fourier-Besselsche Function. Ibid. , vol . 6 9 (1 869) , pp . 1 28-1 41 .
3. Handbuch der Kugelfunctionen. Vols
. I , II . 2 d edition . Berlin , 1 878 , 1 881 .
HELLY, E .
*1. Uber lineare Funktionaloperationen. Sitzungsbericht
e de r mathematisch-natur wissenschaftlichen Klass e de r Akademi e i n Wien , Abteilun g Ha , vol . 1 2 1 (1 91 2) ,
pp. 265-297 .
HERMITE, C .
1. Sur les polyndmes de Legendre. Rendicont
vol. 4 (1 890) , pp . 1 46-1 52 . Oeuvres. Vol
i de l Circol o Matematic o d i Palermo ,
. 4 , pp . 31 4-320 .
406
2. Sur les polyndmes de Legendre. Journa l fu r di e rein e un d angewandt e Mathe matik, vol . 1 0 7 (1 891 ) , pp . 80-83 . Oeuvres. Vol . 4 , pp . 321 -326* .
3. Sur les racines de la fonction spherique de seconde espece. Annale s d e l a Facult y
des Science s d e Toulouse , vol . 4 (1 890) , 1 0 pp . Oeuvres. Vol . 4 , pp . 327-336 .
H E R M I T E , C , an d STIELTJES , T . J .
1. Correspondance d'Hermite
et de Stieltjes. Vols
. I , II . Paris , 1 905 .
HILB, E .
1. Uber die Laplacesche Reihe.
vol. 8 (1 920) , pp . 79-90 .
Mathematisch
e Zeitschrift , vol . 5 (1 91 9) , pp . 1 7-25 ;
HILBERT, D .
1. Ober die Discriminate der
im Endlichen abbrechenden hypergeometrischen Reihe.
Journal fu r di e rein e un d angewandt e Mathematik , vol.1 0 3 (1 888) , pp . 337-345 .
H I L B E R T , D . , an d COURANT , R .
1. Methoden der mathematischen Physik.
HILDEBRANDT, T . H
Vol
. 1 . 2 d edition . Berlin , 1 931 .
.
*1. On integrals related to and extensions of the L'ebesgue integrals. Bulleti
American Mathematica l Society , vol . 2 4 (1 91 8) , pp . 1 1 3-1 44 , 1 77-202 .
n o f th e
HILLE, E .
1. A class of reciprocal functions. Annal s o f Mathematics , (2) , vol . 2 7 (1 926) , pp .
427-464.
2. On Laguerre's series. I , II , III . Proceeding s o f th e Nationa l Academ y o f Sci ences, vol . 1 2 (1 926) , pp . 261 -265 , 265-269 , 348-352 .
3. Bemerkung zu einer Arbeit des Herrn Miintz. Mathematisch e Zeitschrift , vol . 3 2
(1930), pp . 422-425 .
4. Vber die Nullstellen der Hermiteschen Polynome. Jahresberich t de r Deutsche n
Mathematiker-Vereinigung, vol . 4 4 (1 933) , pp . 1 62-1 65 .
HOBSON, E . W
.
1. The Theory of Spherical and Ellipsoidal Harmonics.
Cambridge
, 1 931 .
HOLL6, A .
1. A mechanikus quadraturdrdl. Thesis
. Budapest , 1 939 , 2 3 pp .
JACKSON, D .
1. On functions of closest approximation. Transaction
s of th e American Mathematica l
Society, vol . 2 2 (1 921 ) , pp . 1 1 7-1 28 .
2. Note on a class of polynomials of approximation. Ibid.
, vol . 2 2 (1 921 ) , pp . 320-326 .
3. A generalized problem in weighted approximation. Ibid.
, vol . 2 6 (1 924) , pp . 1 33-1 54 .
4. The Theory of Approximation. America
n Mathematical Societ y Colloquiu m Publi cations, vol . 1 1 , 1 930 .
5. Series of orthogonal polynomials. Annal
s o f Mathematics , (2) , vol . 3 4 (1 933) , pp .
527-545.
6. Certain problems of closest approximation. Bulleti
n o f th e America n Mathematica l
Society, vol . 3 9 (1 933) , pp . 889-906 .
7. The summation of series of orthogonal polynomials. Ibid. , vol . 40 (1 934) , pp. 743-752.
8. Formal properties of orthogonal polynomials in two variables. Duk e Mathematica l
Journal, vol . 2 (1 936) , pp . 423-434 .
JACOB, M . M
.
1. Sullo sviluppo di una funzione di ripartizione in serie di polinomi di Hermite.
Giornale dell'Istitut o Italian o degl i Attuari , vol . 2 (1 931 ) , pp . 1 00-1 06 , 356-368 .
2. Sur le phenomene de Gibbs dans les developpements de series de polynomes d'Hermite.
Comptes Rendu s d e TAcadSmi e de s Sciences , Paris , vol . 20 4 (1 937) , pp . 1 540 1543.
JACOBI, C . G . J .
1. Ueber Gauss 1 neue Methode t die Werthe der Integrale ndherungsweise zu finden.
Journal fu r di e rein e un d angewandt e Mathematik , vol . 1 (1 826) , pp . 301 -308 .
Gesammelte Werke. Vol . 6 , pp . 3-1 1 .
407
2. tjber die Entwickelung des Ausdrucks (aa — 2aa r [co s w co s <f> -f - si n c a si n <f>
•cos ( 0 - 0') ] + a'a')~ 1 /2 Ibid. , vol . 26 , 1 843 , pp . 81 -87 . Gesammelte Werke.
Vol. 6 , pp . 1 48-1 55 .
3. Untersuchungenuber die Differentialgleichung der hypergeometrischen Reihe. Ibid. ,
vol. 5 6 (1 859) , pp . 1 49-1 65 . Gesammelte Werke. Vol . 6 , pp . 1 84-202 .
JORDAN, CAMILLE .
1. Cours d'Analyse de
I'ficole Poly technique. Vol
. 3 , 2 d edition . Paris , 1 896 .
JORDAN, CHARLES .
1. Sur une serie de polynomes dont chaque somme partielle represente la meilleure
approximation d'un degre donne suivant la methode des moindres car res. Pro ceedings o f th e Londo n Mathematica l Society , (2) , vol . 2 0 (1 920) , pp . 297-325 .
JULIA, G .
1. Sur les polynomes de Tchebichef. Compte s Rendu s d e TAcademi e de s Sciences ,
Paris, vol . 1 8 2 (1 926) , pp . 1 201 -1 202 .
KACZMARZ, ST. , an d STEINHAUS , H .
*1. Theorie der Orthogonalreihen. Warszawa-Lw6w
, 1 935 .
KELDYSCH, M. , an d LAVRENTIEFP , M .
1. Sur la representation conforme des domaines limites par des courbes rectifiables.
Annales Scientifique s d e l'Ecole Normal e Superieure , (3) , vol. 5 4 (1 937) , pp . 1 -38 .
KLEIN, F .
1. Ueber die Nullstellen der hypergeometrischen Reihe. Mathematisch e Annalen ,
vol. 3 7 (1 890) , pp . 573-590 . Gesammelte Abhandlungen. Vol . 2 , pp . 550-567 .
KOGBETLIANTZ, E .
1. Sur les series de fonctions ultraspheriques. Compte
s Rendu s d e TAcademi e de s
Sciences, Paris , vol . 1 6 3 (1 91 6) , pp . 601 -603 .
2. Sur la sommation des series ultraspheriques. Ibid.
, vol . 1 6 4 (1 91 7) , pp . 51 0-51 3 ,
626-628, 778-780 ; vol . 1 6 9 (1 91 9) , pp . 54-57 .
3. Sur les developpements de Jacobi. Ibid. , vol . 1 6 8 (1 91 9) , pp . 992-994 .
4. Sur les series ultraspheriques. Ibid.
, vol . 1 6 9 (1 91 9) , pp . 322-324 .
5. Nouvelle observations sur les series ultraspheriques. Ibid.
, vol . 1 6 9 (1 91 9) , pp .
423-426.
6. Sur Vunicite des developpements ultraspheriques. Ibid.
, vol . 1 6 9 (1 91 9) , pp . 769 770, 950-953 .
7. Sur les developpements de Jacobi. Ibid. , vol . 1 7 2 (1 921 ) , pp . 1 333-1 334 ; vol . 1 9 2
(1931), pp . 91 5-91 8 .
8. Sur la sommabilit'e (C , 5) de developpements suivant les polynomes d'Hermite. Ibid. ,
vol. 1 9 2 (1 931 ) , pp . 662-663 .
9. Nouvelles observations sur le systeme orthogonal de polynomes d'Hermite. Ibid. , vol .
192 (1 931 ) , pp . 1 696-1 698 .
10. Sur les series d'Hermite et de Laguerre. Ibid. , vol . 1 9 3 (1 931 ) , pp . 386-389 .
11. Sur la convergence des series d'Hermite. Ibid. , vol . 1 9 4 (1 932) , pp . 1 61 -1 63 .
12. Sur les developpements de Laguerre. Ibid. , vol . 1 9 4 (1 932) , pp . 1 422-1 424 .
13. Sur la serie de Laguerre. Ibid. , vol . 1 9 6 (1 933) , pp . 523-525 .
14. Expression approchee du polynome de Laguerre L^ a ) (x). Ibid. , vol . 1 9 6 (1 933) ,
pp. 1 079-1 080 .
15. Sur la determination du saut D(x 0) def(x). Ibid.
, vo L 1 9 6 (1 933) , pp . 464-466 .
16. fiber die ( C b)-Summierbarkeit der Laplaceschen Reihe fur 1 / 2 < 5 < 1 . Mathema tische Zeitschrift , vol . 1 4 (1922), pp . 99-109.
17. Analogie entre les series trigonometriques et les series spheriques au point de vue dt
leur sommahilite par les moyennes arithmetiques. These . Paris , 1 923 , 6 5 pp .
18. Sur la sommabilite de la serie ultraspherique a Vinterieur de Vintervalle (—1 , + 1 )
par la methode des moyennes arithmetiques. Bulleti n d e l a Societ e Mathematiqu e
de France , vol . 5 1 (1 923) , pp . 244-295 .
408
19. Recherches sur la sommabilite des series ultraspheriques par la methode des moyennes
arithmetiques. Journa l d e Mathematiques , (9) , vol . 3 (1 924) , pp . 1 07-1 87 .
20. Recherches sur Vunicite kdes series ultraspheriques. Tbid.
, (9) , vol . 5 (1 926) , pp .
125-196.
21. Sommation des series et integrates divergentes par les moyennes arithmetiques et
typiques. Memoria l de s Science s Mathematiques . Vol . 51 . Paris , 1 931 .
22. Recherches sur la sommabilite des series d'Hermite. Annale s Scientifique s d e l'ficol e
Normale Superieure , (3) , vol . 4 9 (1 932) , pp . 1 37-221 .
23. Sur les moyennes arithmetiques des series-noyaux des developpements en series d'Hermite et de Laguerre et sur celles de ces series-noyaux derivees terme a terme. Journa l
of Mathematic s an d Physics , Massachusett s Institut e o f Technology , vol . 1 4
(1935), pp . 37-99 .
24. Contribution a Vetude du saut d'une fonction donnee par son developpement en series
d' Hermite ou de Laguerre. Transaction s o f th e American Mathematical Society ,
vol. 3 8 (1 935) , pp . 1 0-47 .
KOROUS, J .
1. O rozvoji funkci jedne redlne promenne v radu Hermiteovych polynomu. Rozprav y
Ceske Akademie , (2) , vol . 3 7 (1 928) , no . 1 1 , 3 4 pp .
2. O faddch Laguerrovych polynomu (wit h Frenc h abstract) . Ibid. , no . 40 , 2 3 pp.
3. O rozvoji funkci jedne redlne promenn'e v radu jistych ortogondlnich polynomk (wit h
English abstract) . Ibid. , vol . 4 8 (1 938) , 1 2 pp .
4. Vber Reihenentwicklungen nach
verallgemeinerten Laguerreschen Polynomen mil
drei Parametern. Vestni k Kr&lovsk e Cesk e Spolecnost i Naiik , THd a Matemat. Pffrodoved, 1 937 , 2 6 pp .
5. Vber Entwicklungen der Funktionen einer reellen Veranderlichen in Reihen einer
gewissen Klasse orthogonaler Polynome im unendlichen Intervalle. Ibid. , 1 937 ,
19 pp .
KOSCHMIEDER, L .
1. tfber besondere Jacobische Polynome.
pp. 1 23-1 37 .
Mathematisch
e Zeitschrift , vol . 8 (1 920) ,
KOWALBWSKI, G .
*1. Einfuhrung in die Determinantentheorie. Leipzig
, 1 909 .
KOWALLIK, U .
1. Entwicklung einer willkurlichen Funktion nach Hermiteschen Orthogonalfunktionen.
Mathematische Zeitschrift , vol . 3 1 (1 930) , pp . 498-51 8 .
KRALL, H . L .
1. On derivatives of orthogonal polynomials. Bulleti n o f th e America n Mathematica l
Society, vol . 4 2 (1 936) , pp . 423-428 .
2. On higher derivatives of orthogonal polynomials. Ibid.
, vol . 4 2 (1 936) , pp . 867-870 .
KRAWTCHOUK, M .
1. Sur une generalisation des polynomes d'Hermite. Compte s Rendu s d e TAcademi e
des Sciences , Paris , vol . 1 8 9 (1 929) , pp . 620-622 .
2. Sur la distribution des racines des polynomes orthogonaux. Ibid. , vol . 1 9 6 (1 933) ,
pp. 739-741 .
KRONECKER, L .
*1. Zur Theorie der Elimination einer Variabeln aus zwei algebraischen Gleichungen.
Monatsberichte de r Preussische n Akademi e de r Wissenschafte n z u Berlin , 1 881 ,
pp. 535-600 .
LAGRANGE, J . L .
1. Oeuvres. Vol
. 1 , pp . 534r-539 .1 867 .
LAGUERRE, E . N .
1. Sur Vintegrate f * x ~ l e ~ x dx. Bulleti n d e l a Societ 6 Mathematiqu e d e France ,
vol. 7 (1 879) , pp . 72-81 . Oeuvres. Vol . 1 , pp . 428-437 .
409
2. Sur Vapproximation des fonctions circulaires au moyen des fonctions algkbriques.
Comptes Rendu s d e l'Academi e de s Sciences , Paris , vol . 9 0 (1 880) , pp . 304r-307 .
Oeuvres. Vol . 1 , pp . 1 04-1 07 .
3. Sur les Equations alg'ebriques dont le premier membre satisfait a une Equation
lineaire du second ordre. Ibid. , vol . 9 0 (1 880) , pp . 809-81 2 . Oeuvres, Vol . 1 ,
pp. 1 26-1 32 .
LANGBR, R . E .
*1. The asymptotic solutions of ordinary linear differential equations of the second order,
with special reference to the Stokes phenomenon. Bulleti n o f th e America n
Mathematical Society , vol . 4 0 (1 934) , pp . 546-582 .
*2. The asymptotic solutions of certain linear ordinary differential equations of the
second order. Transaction s o f th e America n Mathematica l Society , vol . 3 6
(1934), pp . 90-1 06 .
*3. On the asymptotic solutions of ordinary differential equations, with reference to the
Stokesf phenomenon about a singular point. Ibid. , vol . 3 7 (1 935) , pp . 397-41 6 .
LAWTON, W .
1. On the zeros of certain polynomials related to Jacobi and Laguerre polynomials.
Bulletin o f th e America n Mathematica l Society , vol . 38 , (1 932) , pp . 442-448 .
LEBESGUE, H .
*1. Sur la divergence et la convergence non-uniforme des series de Fourier. Compte
Rendus, d e l'Academi e de s Sciences , Paris , vol . 1 4 1 (1 905) , pp . 875-877 .
*2. Lecons sur les Series TrigonomUriques. Paris
, 1 906 .
s
LEGENDRE, A . M .
1. Exercices de Calcul Integral sur Divers Ordres de Transcendantes. Vol
. 2 . Paris ,
1817.
L E ROY , E .
1. Sur les series divergentes et les fonctions definies par un developpement de Taylor.
Annales d e l a Facult e de s Science s d e Toulouse , (2) , vol . 2 (1 900) , pp . 31 7-430 .
LTJKXCS, F .
1. Verscharfung des ersten Mittelwertsatzes der Integralrechnung fur rationale Polynome. Mathematisch e Zeitschrift , vol . 2 (1 91 8) , pp . 295-305 .
2. fiber die Laplacesche Reihe. Ibid. , vol . 1 4 (1 922) , pp . 250-262 .
MARCINKIEWICZ, J .
*1. Quelques remarques sur Vinterpolation. Act
a Litteraru m a c Scientiaru m Regia e
Universitatis Hungarica e Francisco-Jdsephinae , vol . 8 (1 937) , pp . 1 27-1 30 .
2.» Sur la divergence des polynomes d 1 interpolation. Ibid.
, vol . 8 (1 937) , pp . 1 31 -1 35 .
MARKOFF, A .
1. On some applications of algebraic continued fractions (i n Russian) . Thesis . St
Petersburg, 1 884 , 1 3 1 pp .
2. Demonstration de certaines in&galit&s de M. Tchebycheff. Mathematisch e Annalen
vol. 2 4 (1 884) , pp . 1 72-1 80 .
3. Extrait d'une lettre. Annale s Scientifique s d e Pficol e Normal e Superieure , (3)
vol. 2 (1 885) , p . 1 83 .
4. Sur les racines de certaines equations (second note). Mathematisch e Annalen
vol. 2 7 (1 886) , pp . 1 77-1 82 .
5. Differenzenrechnung. Leipzig
, 1 896 .
.
,
,
,
MEHLER, F . G .
1. Bemerkungen zur Theorie der mechanischen Quadraturen. Journa l fu r di e rein e
und angewandt e Mathematik , vol . 6 3 (1 864) , pp . 1 52-1 57 .
2. Ueber die Entwicklung einer Function von beliebig vielen Variablen nach Laplaceschen Functionen hbherer Ordnung. Ibid. , vol . 6 6 (1 866) , pp . 1 61 -1 76 .
410
3. U Keber die Vertheilung der statischen Elektricitdt in einem von zwei Kugelkalotten
begrenzten Korper. Ibid. , vol . 68 (1868), pp. 134-150.
4. Ueber die Darstellung einer willkilrlichen Function zweier Varidblen durch Cylinderfunctionen. Mathematisch e Annalen , vol . 5 (1 872) , pp . 1 35-1 40 .
5. Notiz uber die DirichleV schen Integralausdrucke filr die Kugelfunction P n (cos 0 )
und uber eine analoge Integralform fiiir die Cylinderfunction J(x). Ibid. , vol . 5
(1872), pp. 1 41 -1 44 .
MEIXNER, J .
1. Orthogonale Polynomsysteme mit einer besonderen Gestalt der erzeugenden Funklion. Journa l o f th e Londo n Mathematica l Society , vol . 9 (1 934) , pp . 6-1 3 .
2. Erzeugende Funktionen der Charlierschen Polynome. Mathematisch e Zeitschrift ,
vol. 4 4 (1 938) , pp . 531 -535 .
MOECKLIN, E .
1. Asymptotische Entwicklungen der Laguerreschen Polynome. Commentari i Mathe matici Helvetici, vol. 7 (1934), pp. 24-46.
MUNTZ, C .
1. tfber die Potenzsummation einer Entwicklung nach Hermiteschen Polynomen.
Mathematische Zeitschrift , vol . 3 1 (1929), pp. 350-355.
MYLLER-LEBEDEFF, V .
1. Die Thporie der Integralgleichungen in Anwendung auf einige Reihenentwicklungen.
Mathematische Annalen , vol . 6 4 (1907), pp. 388-416.
NEUMANN, E . R .
1. Die Entwicklung willkurlicher Funktionen nach den Hermiteschen und Laguerreschen
Orthogonalfunktionen auf Grund der Theorie der Integralgleichungen. Inaugural Dissertation. Breslau , 1 91 2 .
2. Beitrdge zur Kenntnis der Laguerreschen Polynome. Jahresberich t de r Deutsche n
Mathematiker-Vereinigung, vol . 3 0 (1 921 ) , pp . 1 5-35 .
OBRECHKOFF, N .
1. Sur la sommation de la serie ultraspherique par la mUhode des moyennes arithmktiques. Rendicont i de l Circol o Matematic o d i Palermo , vol . 5 9 (1 936) , pp .
266-287.
2. Formules asymptotiques pour les polynomes de Jacobi et sur les series suivant les
mbmes polynomes. Annuair e d e PUniversit e d e Sofia , Facult e Physico-Mathe matique, vol . 1 (1936), pp. 39-133.
PERRON, O .
1. Uber das infinitdre Verhalten der Koeffizienten einer gewissen Potenzreihe. Archi v
der Mathemati k un d Physik , (3) , vol . 2 2 (1 91 4) , pp . 329-340 .
2. Uber das Verhalten einer ausgearteten hypergeometrischen Reihe bei unbegrenztem
Wachstum eines Parameters. Journa l fu r di e rein e un d angewandt e Mathe matik, vol . 1 5 1 (1 921 ) , pp . 63-78.
3. Die Lehre von den Kettenbriichen. 2 d edition . Leipzig , 1 929 .
*4. Algebra. Vols . I , II , 2 d edition. Berlin , 1 932 , 1933.
PLANCHEREL, M. , an d ROTACH , W .
1. Sur les valeurs asymptotiques des polynomes d fHermite H n(x) = (—l) ne* l2
• dn(e-*2h)/dxn. Commentari i Mathematic i Helvetici , vol . 1 (1929), pp. 227-254.
P6LYA, G .
1. Sur un Moreme de Stieltjes. Compte s Rendu s d e TAcad6mie de s Sciences , Paris ,
vol. 1 5 5 (1 91 2) , pp . 767-769 .
2. Sur un algorithme toujours convergent pour obtenir les polynomes de meillcure approximation de Tchebychef pour une fonction continue quelconque. Ibid. , vol . 157
(1913), pp. 840-843.
LIST
1
O F REFERENCE S 4
1
*3. Ober die Nullstellen gewisser ganzer Funktionen. Mathematisch
e Zeitschrift , vol .
2*(1918), pp . 352-383 .
4. Uber die Konvergenz von Quadraturverfahren. Ibid.
, vol . 3 7 (1 933) , pp . 264-286 .
P 6 L Y A , G. , an d SZEGO , G .
1. Aufgaben und Lehrsdtze aus der Analysis. Vols
. I , II . Berlin , 1 925 .
2. Uber den transfiniten Durchmesser (Kapazitatskonstante) von
ebenen und rdumlichen
Punktmengen. Journa l fu r di e rein e un d angewandt e Mathematik , vol . 1 6 5
U931), pp . 4-49 .
POPOVICIU, T .
1. Sur la distribution des
zeros de certains polynomes minimisants. Bulleti
nd e
l'Academie Roumaine , vol . 1 6 (1 934) , pp . 21 4-21 7 .
2. Sur certains problemes de maximum de Stieltjes. Bulleti n M a t h e m a t i q u e d e l a
Societ6 Roumain e de s Sciences , vol . 3 8 (1 936) , pp . 7 3 96 .
RAU, H .
1. Uber die Lebesgueschen Konstanten der
Reihenentwicklungen nach
Jacobischen
Polynomen. Journa l fu r di e rein e un d angewandt e Mathematik , vol . 1 6 1 (1 929) ,
pp. 237-254 .
2. Uber eine asyrnptotische Darstellung der Jacobischen Polynome durch Besselsche
Funktionen. Mathematisch e Zeitschrift , vol . 4 0 (1 936) , pp..683-692 .
RIESZ, F .
*1. Sur certains systemes singuliers d }equations integrates. Annale s Scientifique s d e
l'Ecole Normal e Superieure , (3) , vol . 2 8 (1 91 1 ) , pp . 33-62 .
*2. Ober die Randwerte einer analytischen Funktion.
Mathematisch
e Zeitschrift ,
vol. 1 8 (1 923) , pp . 87-95 .
RIESZ, M .
1. Eine trigonometrische Interpolationsformel und
einige Ungleichungen fur Polynome.
Jahresbericht de r Deutsche n Mathematiker-Vereinigung , vol . 2 3 (1 91 5) , pp .
354-368.
2. Sur le probleme des moments. Arki v fo r Matematik , Astronom i oc h Fysik , vol . 1 6
(1921), no . 1 2 , 2 3 pp. ; vol . 1 6 (1 922) , no . 1 9 , 2 1 pp. ; vol . 1 7 (1 923) , no . 1 6 , 5 2 pp .
ROTACH, W .
1. Reihenentwicklungen einer
willkiirlichen Funktion nach Hermiteschen und Laguerreschen Polynomen. Inauguraldissertation , Eidgenossisch e Technisch e Hoch schule Zurich , 1 925 .
SCHMIDT, E .
1. Ober die Charlier-Jordansche Entwicklung
einer willkiirlichen Funktion nach der
Poissonschen Funktion
und
ihren Ableitungen. Zeitschrif
t fii r angewandt e
Mathematik un d Mechanik , vol . 1 3 (1 933) , pp . 1 39-1 42 .
SCHUR, I .
1. Ober die Verteilung der Wurzeln bei gewissen algebraischen Gleichungen mil ganzzahligen Koeffizienten. Mathematisch e Zeitschrift , vol . 1 (1 91 8) , pp . 377-402 .
2. Affektlose Gleichungen in der Theorie der Laguerreschen und Hermiteschen Polynome. Journa l fu r di e rein e un d angewandt e Mathematik , vol . 1 6 5 (1 931 ) ,
pp. 52-58 .
SCHWID, N .
1. The asymptotic forms of the Her mite and Weber functions. Transaction
American Mathematica l Society , vol . 3 7 (1 935) , pp . 339-362 .
s o f th e
S E N , D . N. , an d RANGACHARIAR , V .
1. Generalized Jacobi polynomials. Bulleti
vol. 4 2 (1 936) , pp . 901 -908 .
n o f th e America n Mathematica l Society ,
SHERMAN, J .
1. On the numerators of the convergents of the Stieltjes continued fractions. T r a n s
actions o f th e America n Mathematica l Society , vol . 3 5 (1 933) , pp . 64-87 .
-
4 1 2 LIS
T O F REFERENCE S
SHIBAT A, K .
1. On the distribution of the roots of a polynomial satisfying a certain differential equation of the second order. Japanes e Journa l o f Mathematics , vol . 1 (1 924) , p p .
147-153.
SHOHAT, J .
1. On the-polynomial of the best approximation to a given continuous function. Bulleti
n
of t h e America n Mathematica l Society , vol . 3 1 (1 925) , p p . 509-51 4 .
2. On a general formula in the theory of Tchebycheff polynomials and its applications.
Transactions o f th e America n Mathematica l Society , vol . 2 9 (1 927) , p p . 569-583 .
3. On a certain formula of mechanical quadratures with non-equidistant ordinates.
Ibid., vol . 3 1 (1 929) , p p . 448-463 .
4. On the polynomial and trigonometric approximation of measurable bounded functions on a finite interval. Mathematisch e Annalen , vol . 1 0 2 (1 929) , p p . 1 57-1 75 .
5. On interpolation. Annal
s o f Mathematics , (2) , vol . 3 4 (1 933) , p p . 1 30-1 46 .
6. Theorie Generate des Polynomes Orthogonaux de Tchebichef. Memoria l de s Science s
M a t h e m a t i q u e s . V o l . 6 6 . Paris , 1 934 .
7. On mechanical quadratures, in particular, with positive coefficients. Transaction s
of t h e America n M a t h e m a t i c a l Society , vol . 4 2 (1 937) , p p . 461 -496 .
8. On the convergence properties of Lagrange interpolation based on the zeros of orthogonal
Tchebycheff polynomials. Annal
s o f Mathematics , (2) , vol . 38 (1 937) , pp . 758-769.
SMIRNOFF, V . J . '
1. Sur la th&orie des polynomes orthogonaux a une variable complexe. Journa l d e l a
Societe Physico-Mathematiqu e d e Leningrad , vol . 2 (1 928) , p p . 1 55-1 79 .
2. Sur les formules de Cauchy et de Green et quelques problemes qui s'y rattachent.
Bulletin d e TAcademi e de s Science s d e l'-URSS , 1 932 , pp . 337-372 .
SMITH, E . R
.
1. Zeros of the Hermilian polynomials.
(1936), pp . 354-358 .
America
n Mathematica l Monthly , vol . 4 3
SONIN(E), N . J .
1. Recherches sur les fonctions cylindriques et le developpement des fonctions continues
en series. Mathematisch e Annalen , vol . 1 6 (1 880) , p p . 1 -80 .
2. On the precision of the determination of limiting values of definite integrals (i n Rus sian). Zapisk i Akademi i N a u k , vol . 6 9 (1 892) , p p . 1 -30 .
SPENCER, V . E .
1. Asymptotic expressions for the zeros of generalized Laguerre polynomials and
functions. D u k
e Mathematica l Journal , vol . 3 (1 937) , p p . 667-675 .
Weber
STEKLOFF, W .
1. Sur les expressions asymptotiques de certaines fonctions, definies par les equations
differentiates lineaires
du second order, et leurs applications au probleme du
developpement d }une fonction arbitraire en series procedant suivant les-dites fonctions. Communication s d e l a Soeiet 6 M a t h e m a t i q u e d e Kharkow , (2) , vol . 1 0
(1907), p p . 97-200 . Remarque complementaire, p . 201 .
2. On approximate evaluation of definite integrals by means of formulas of mechanical
quadratures. I . Convergence of formulas of mechanical quadratures (i n Russian) .
Bulletin d e TAcademi e Imperial e de s Sciences , Petrograd , (6) , vol . 1 0 (1 91 6) ,
p p . 1 69-1 86 .
STIELTJES, T . J
.
1. Quelques recherches sur la theorie des quadratures dites mkchaniques. Annale s
Scientifiques d e l'ficol e Normal e Superieure , (3) , vol . 1 (1 884) , p p . 409-426 .
Oeuvres Completes. Vol . 1 , p p . 377-394 .
2. Note a Voccasion de la reclamation de M. Markoff. Ibid. , (3) , vol . 2 (1 885) , pp .
183-184. Oeuvres Completes. Vol . 1 , p p . 430-431 .
4.1 3
3. Sur certains polynomes qui verifient une equation differ entielle lineaire du second
ordre et sur la theorie des functions de Lame. Act a Mathematica , vol . 6 (1 885) ,
p p . 321 -326 . Oeuvres Completes. Vol . 1 , pp . 434-439 .
4. Sur quelques theorbmes d'algebre. Compte s Rendu s d e l'Academi e de s Sciences ,
Paris, vol . 1 0 0 (1 885) , pp . 439-440 . Oeuvres Completes. Vol . 1 , p p . 440-441 .
5. Sur les polynomes de Jacobi. Compte s Rendu s d e TAcad6mi e de s Sciences , Paris ,
vol. 1 0 0 (1 885) , pp . 620-622 . Oeuvres Completes. Vol . 1 , p p . 442-444 .
6. Sur les racines de Vequation X n = 0 . Act a Mathematica , vol . 9 (1 886) , pp . 385-400 .
Oeuvres Completes. Vol . 2 , pp . 73-88 .
7. Sur la valeur asymptotique des
polyndmes de Legendre. Compte s Rendu s d e
l'Academie de s Sciences , Paris , vol . 1 1 0 (1 890) , pp . 1 026-1 027 . Oeuvres Completes. Vol . 2 , p p . 234-235 .
8. Sur les polynomes de Legendre. Annale s d e l a Facult y de s Science s d e Toulouse ,
vol. 4 (1 890) , 1 7 pp . Oeuvres Completes. Vol . 2 , p p . 236-252 .
9. Sur les racines de la fonction spherique de seconde espece. Annale s d e l a Facult e
des Science s d e Toulouse , vol . 4 (1 890) , 1 0 pp . Oeuvres Completes. Vol . 2 ,
pp. 253-262 .
10. Recherches sur les fractions continues.
Compte s Rendu s d e l'Academi e de s
Sciences, Paris , vol . 1 1 8 (1 894) , pp . 1 401 -1 403 . Oeuvres Completes. Vol . 2 , p p .
398-401.
11. Recherches sur les fractions continues. Annale s d e l a Facult y de s Science s d e
Toulouse, vol . 8 (1 894) , 1 2 2 p p . ; vol . 9 , 1 895 , 4 7 pp . Oeuvres Completes.
Vol. 2 , p p . 402-566 .
12. Sur certaines inegalites dues a M. P. Tchebychef. Articl e r6dig 6 d'apre s u n m a n u script inedit . Oeuvres Completes. Vol . 2 , pp . 586-593 .
STONE, M . H
.
1. Developments in Hermite polynomials. Annal
s o f Mathematics , (2) , vol . 2 9 (1 928) ,
p p . 1 -1 3 .
*2. Linear Transformations in
Hilbert Space and their Applications to
Analysis.
American Mathematica l Societ y Colloquiu m Publications , vol . 1 5 , 1 932 .
SZASZ, O .
*1. Korldtos hatvdnysorokrdl. M a t h e m a t i k a
43 (1 926) , p p . 504-520 .
i e s Term6szettudom&ny i Ertesfto , vol .
SZEGO, G .
1. A Hankel-fele formdkrol. Mathematika i e s Termeszettudomany i Ertesftd , vol
36 (1 91 8) , p p . 497-538 .
2. Ein Beilrag zur Theorie der Polynome von Laguerre und Jacobi. Mathematisch
Zeitschrift, vol . 1 (1 91 8) , p p . 341 -356 .
3. Ober Orthogonalsysteme von Polynomen. Ibid.
, vol . 4 (1 91 9) , p p . 1 39-1 51 .
4. Beitrdge zur Theorie der Toeplitzschen Formen.
I I . Ibid. , vol . 9 (1 921 ) , p p
167-190.
5. Ober orthogonale Polynome, die zu einer gegebenen Kurve der komplexen Ebene
gehbren. Ibid. , vol . 9 (1 921 ) , p p . 21 8-270 .
6. Ober die Entwickelung einer
analytischen Funktion nach den Polynomen eines
Orthogonalsystems. Mathematisch
e Annalen , vol . 82 , 1 921 , pp . 1 88-21 2 .
7. Ober die Randwerte analylischer Funklionen. Ibid.
, vol . 8 4 (1 921 ) , p p . 232-244 .
8. Ober den asymptotischen Ausdruck von Polynomen, die durch cine Orthogonalitdtseigenschaft definiert sind. Ibid. , vol . 8 6 (1 922) , p p . 1 1 4-1 39 .
9. Ober die Entwicklung einer
willkurlichen Funktion
nach den Polynomen eines
Or thogonalsy stems. Mathematisch e Zeitschrift , vol . 1 2 (1 921 ) , p p . 61 -94 .
10. Beitrdge zur Theorie der Laguerreschen Polynome. I
. Entwicklungssdtze. Ibid.
vol. 2 5 (1 926) , p p . 87-1 1 5 .
.
e
.
,
414
11. Bvmerkungen zu einer Arbeit von Herrn Fejer uber die Legendreschen Polynome.
Ibid., vol . 2 5 (1 926) , pp . 1 72-1 87 .
12. Ein Beitrag zur Theorie der Thetafunktionen. Sitzungsbericht
e de r Preussische n
Akademie de r Wissenschaften , physikalish-mathematisch e Klasse , 1 926 , pp .
242-252.
13. Koeffizientenabschdtzungen bei
ebenen und rdumlichen harmonischen Entwicklungen.
Mathcmatische Annalen , vol . 9 6 (1 927) , pp . 601 -632 .
14. Uber gewisse Inter polationspolynome, die
zu den Jacobischen und Laguerreschen
Abszissen gehoren. Mathematisch e Zeitschrift , vol . 3 5 (1 932) , pp . 579-602 .
15. Uber einige asymptotische Entwicklungen der Legendreschen Funklionen. Proceed
ings o f th e Londo n Mathematica l Society , (2) , vol . 3 6 (1 932) , pp . 427-450 .
16. Uber eine von Herrn S. Bernstein herruhrende Abschdtzung der Legendreschen Polynome. xMathematisch e Annalen , vol . 1 0 8 (1 933) , pp . 360-369 .
17. Asymptotische Entwicklungen
der
Jacobischen Polynome. Schrifte n de r Konigs berger Gelehrte n Gesellschaft , naturwissenschaftlich e Klasse , vol . 1 0 (1 933) ,
pp. 35-1 1 2 .
18. Bemerkungen zu einem Satz von E. Schmidt uber algebraische Gleichungen. Sitzungs
berichte de r Preussische n Akademi e de r Wissenschaften , physikalish-mathema tische Klasse , 1 934 , pp . 3-1 5 .
19. Uber gewisse orthogonale Polynome, die zu einer oszillierenden Belegungsfunktion
gehoren. Mathematisch e Annalen , vol . 1 1 0 (1 934) , p p . 501 -51 3 .
20. Inequalities for the zeros of Legendre polynomials and related functions. T r a n s
actions o f th e America n Mathematica l Society , vol . 3 9 (1 936) , pp . 1 -1 7 .
21. An integral equation for the square of a Laguerre polynomial. Journa
l o f th e Londo n
Mathematical Society , vol . 1 2 (1 937) , pp . 1 62-1 63 .
TAMARKIN, J . D .
1. On the Theory of Polynomials of
University, 1 935-1 936 .
Approximation. Lectur
e delivere d a t Brow n
TCKAKALOFF, L .
1. Sur la structure des ensembles lineaires definis par une certaine propriety minimale.
Acta Mathematica , vol . 6 3 (1 934) , pp . 77-97 .
TCHEBICHEF, P . L .
1. Sur les fractions continues. Journa l d e Mathematiques , (2) , vol . 3 (1 858) , pp .
289-323. Oeuvres. Vol . 1 , pp . 201 -230 .
2. Sur Vinterpolation par la methode des moindres carres. Memoire s d e TAcademi c
Imperiale de s Science s d e St . Petersbourg , (7) , vol . 1 (1 859) , pp . 1 -24. Oeuvres.
Vol. 1 , pp . 471 -498 .
3. Sur le developpement des fonctions a une seule variable. Bulleti n d e l'Acad6mi e
Imperiale de s Science s d e St . Petersbourg , vol . 1 (1 859) , pp . 1 93-200 . Oeuvres.
Vol. 1 , pp . 499-508 .
4. Sur Vinterpolation. Zapisk
i Akademi i Nauk , vol . 4 , Supplemen t no . 5 , 1 864 .
Oeuvres. Vol . 1 , pp . 539-560 .
5. Sur les fonctions analogues a celles de Legendre. Zapisk i Akademi i Nauk , vol . 1 6
(1870), pp . 1 31 -1 40 . Oeuvres. Vol . 2 , pp . 59-68 .
6. Sur les valeurs limites des integrates. Journa l d e Mathematiques , (2) , vol . 1 9
(1874), pp . 1 57-1 60 . Oeuvres. Vol . 2 , pp . 1 81 -1 85 .
7. Sur le rapport de deux integrates etendues aux m&mes valeurs de la variable. Za piski Akademi i Nauk , vol . 44 , Supplemen t no . 2 , 1 883 . Oeuvres.. Vol . 2 ,
pp. 375-402 .
8. Sur la representation des valeurs limites des integrates par des residus integraux.
Ibid., vol . 51 , Supplemen t no . 4 , 1 885 . Oeuvres. Vol . 2 , pp . 41 9-440 . Act a
Mathematica, vol . 9 (1 886 ) pp . 35-56 .
415
TlTCHMARSH, E . C .
*1. The Theory of Functions. Oxford
, 1932.
TRICOMI, F .
1. Trasformazione di
Laplace e polinomi di Laguerre. I . Inversions delta trasformazione. Rendicont i dell a R . Accademi a de i Lincei , (6) , vol . 2 1 (1 935) , pp .
232-239.
USPENSKY, J . V .
1. On the development of arbitrary functions in series of Hermite's and Laguerre's
polynomials. Annal
s o f Mathematics , (2) , vol . 2 8 (1 927) , pp . 593-61 9 .
VAN VEEN , S . C .
1. Asymptotische Entwicklung und Nullstellenabsch&tzung der Hermiteschen Funktionen. Proceedings , Koninklijk e Akademi e va n Wetenschappen t e Amsterdam ,
vol. 3 4 (1931), pp . 257-267.
2. Asymptotische Entwicklung und Nullstellenabschatzung der
Hermiteschen Funktionen. Mathematisch e Annalen , vol . 1 0 5 (1 931 ) , pp . 408-436 .
3. Zusatz zum vorangehenden Berichte (o f W . Hahn) . Jahresberich t de r Deutsche n
Mathematiker-Vereinigung, vol . 4 4 (1 934) , pp . 236-238 .
VITALI, G. , an d SANSONE , G .
1. Moderna Teoria delle Funzioni di Variabile Reale.
zioni Ortogonali. Bologna , 1 935 .
II
. Sviluppi in
Serie di Fun-
WALSH, J . L .
1. Interpolation and Approximation by
Rational Functions in the Complex Domain.
American Mathematica l Societ y Colloquiu m Publications , vol . 20 , 1935.
WANGERIN, A .
1. Theorie der Kugelfunktionen und
der verwandten Funktionen, insbesondere der
Lameschen und Besselschen (Theorie spezieller, durch lineare Differentialgleichungen definierter Funktionen). Encyklopadi
e de r Mathematische n Wissen schaften, vol . II.1 .2 , pp . 695-759 .
WATSON, G . N .
1. The harmonic functions associated with the parabolic cylinder. Proceeding s o f th e
London Mathematica l Society , (2) , vol. 8 (1 91 0) , pp . 393-421; (2) , vol. 1 7 (1 91 8) ,
pp. 1 1 6-1 48 .
2. Approximate formulae for Legendre functions. Messenge r o f Mathematics , vol .
47 (1 91 8) , pp . 1 51 -1 60 .
3. A Treatise on the Theory of Bessel Functions. Cambridge , 1 922 .
4. Notes on generating functions of polynomials: 1(
) Laguerre polynomials. Journa
l
of th e Londo n Mathematica l Society , vol . 8 (1 933) , pp . 1 89-1 92 .
5. Notes on generating functions of polynomials: (2 ) Hermite polynomials. Ibid.
,
vol. 8 (1 933) , pp . 1 94-1 99 .
6. Vber eine Reihe aus verallgemeinerten Laguerre'schen Polynomen. Sitzungsbericht
e
der mathematisch-naturwissenschaftliche n Klass e de r Akademi e Wien , I la,
vol. 1 4 7 (1 938) , pp . 1 51 -1 59 .
WEYL, H .
1. Singuldre Integralgleichungen mit besonderer Beriicksichtigung des Fourier schen
Integraltheorems. Inauguraldissertation
, Gottingen , 1 908 . Mathematisch e An nalen, vol . 6 6 (1 909) , pp . 273-324 .
WHITTAKER, E . T. , AN D WATSON, G . N .
1. A Course of Modern Analysis. 4t
h edition . Cambridge , 1 935 .
WIGERT, S .
1. Contributions a la theorie des polynomes d'Abel-Laguerre. Arki
v fo r Matematik ,
Astronomi oc h Fysik , vol . 1 5 (1 921 ) , no . 25, 22 pp.
2. Sur les polynomes orthogonaux el Vapproximation des fonctions continues. Ibid. ,
vol. 1 7 (1923), no . 18, 1 5 pp.
416
FURTHER REFERENCE S
WlMAN, A .
1. Uber eine asymptotische Eigenschaft der Ableitungen der ganzen Funktionen von den
Geschlechtern 1 und 2 mit einer endlichen Anzahl von Nullstellen. Mathematisch e
Annalen, vol . 1 0 4 (1 931 ) , pp . 1 69-1 81 .
WINSTON, C .
1. On mechanical quadratures formulae involving the classical orthogonal polynomials.
Annals o f Mathematics , (2) , vol . 3 5 (1 934) , pp . 658-677 .
WRIGHT, E . M .
1. The coefficients of a certain power series. Journa l o f th e Londo n Mathematica l
Society, vol . 7 (1932), pp. 256-262.
YOUNG, W . H .
1. On the connexion between Legendre series and Fourier Series. Proceeding s o f th e
London Mathematical Society , (2) , vol . 1 8 (1 91 9) , pp . 1 41 -1 62 .
ZERNIKE, F .
1. Eine asymptotische Entwicklung fur die grosste Nullstelle der Hermiteschen Polynome.
Koninklijke Akademi e va n Wetenschappe n t e Amsterdam , Proceedings , vol . 34
(1931), pp . 673-680.
ZYGMUND, A .
1. Sur la theorie riemannienne de certains syslemes orthogonaux, II . Prac e Matema tyczno-Fizyczne, vol . 3 9 (1 932) , pp . 73-1 1 7 .
2. Trigonometrical Series. Warszawa-Lw6w , 1 935 .
FURTHER REFERENCE S
ACHIESER, N. I. (See List of References above.)
3. Theory of Approximation. Ne w York , 1 956 .
4. The Classical Moment Problem. Oliver an d Boyd , Edinburg h an d London , 1 965 .
ASKEY, R .
1. Orthogonal expansions with positive coefficients. Proceeding s o f th e America n Mathe matical Society , vol . 1 6 (1 965) , pp . 1 1 91 -1 1 94 .
2. Jacobi polynomial expansions with positive coefficients and imbeddings of projective
spaces. Bulletin o f the American Mathematica l Society , vol . 7 4 (1 968) , pp . 301 -304 .
3. A transplantation theorem for Jacobi series. Illinoi s Journa l o f Mathematics , vol . 1 3
(1969), pp . 583-590.
4. Linearization of the product of orthogonal polynomials, in Problems in analysis. Edite d
by R . Gunning , Princeto n Universit y Press , Princeton , N . J. , 1 970 , pp . 223-228 .
5. Positivity of the Cotes numbers for some Jacobi abscissas. Numerisch e Mathematik ,
vol. 1 9 (1 972) , pp . 46-48 .
6. Mean convergence of orthogonal series and Lagrange interpolation. Acta Maihematic a
Academiae Scientiaru m Hungaricae , vol . 2 3 (1 972) , pp . 71 -85 .
7. Grunbaum's inequality for Bessel functions. Journa l o f Mathematica l Analysi s an d
Applications, vol . 4 1 (1 973) , pp . 1 22-1 24 .
8. Summability of Jacobi series. Transaction s o f th e America n Mathematica l Society ,
vol. 1 7 9 (1 973) , pp . 71-84.
9. Certain rational functions whose power series have positive coefficients. II . SIA M
Journal o n Mathematica l Analysis , vol . 5 (1 974) , pp . 53-57 .
10. Jacobi polynomials. I. New proofs of Koornwinder's Laplace type integral representation
and Bateman's bilinear sum. SIAM Journa l on Mathematical Analysis , vol . 5 (1974),
pp. 1 1 9-1 24 .
ASKEY, R. , AN D FITCH , J .
1. Positivity of the Cotes numbers for some ultraspherical abscissas. SIA M Journa l o n
Numerical Analysis , vol . 5 (1 968) , pp . 1 99-201 .
2. Integral representations for Jacobi polynomials and some applications. Journa l o f
Mathematical Analysi s an d Applications , vol . 2 6 (1 969) , pp . 41 1 -437 .
FURTHER REFERENCE S
417
ASKEY, R. , AN D GASPER , G .
1. Linearization of the product of Jacobi polynomials. III . Canadia n Journa l o f Mathe matics, vol . 2 3 (1 971 ) , pp . 332-338 .
2. Jacobi polynomial expansions of , Jacobi polynomials with non-negative coefficients.
Proceedings o f th e Cambridg e Philosophica l Society , vol . 7 0 (1 971 ) , pp . 243-255 .
3. Certain rational functions whose power series have positive coefficients. America n
Mathematical Monthly , vol . 7 9 (1 972) , pp . 327-341 .
4. Positive Jacobi polynomial sums. II . America n Journa l o f Mathematics .
ASKEY, R. , AN D POLLARD , H .
1. Some absolutely monotonic and completely monotonic functions. S I AM Journa l o n
Mathematical Analysis , vol . 5 (1 974) , pp . 58-63 .
ASKEY, R. , AN D STEINIG, J .
1. Some positive trigonometric sums. Transaction s o f th e America n Mathematica l
Society, vol . 1 8 7 (1 974) , pp . 295-307 .
2. A monotonic trigonometric sum. America n Journa l o f Mathematics .
ASKEY, R. , AN D WAINGER, S .
1. Mean convergence of expansions in Laguerre and Hermite series. America n Journa l
of Mathematics , vol . 8 7 (1 965) , pp . 695-708 .
2. A transplantation theorem between ultraspherical series. Illinois Journal o f Mathematics ,
vol. 1 0 (1 966) , pp . 322-344 .
3. A transplantation theorem for ultraspherical coefficients. Pacific Journa l o f Mathe matics, vol . 1 6 (1 966) , pp . 393-405 .
4. A convolution structure for Jacobi series. America n Journa l o f Mathematics , vol . 9 1
(1969), pp . 463-485.
BAILEY, W . N .
1. The generating function of Jacobi polynomials. Journa l o f th e Londo n Mathematica l
Society, vol . 1 3 (1 938) , pp . 8-1 2 .
2. On the product of two Laguerre polynomials. Quarterl y Journa l o f Mathematics ,
vol. 1 0 (1 939) , pp . 60-66 .
BALAZS, J. , AN D TURAN, P .
1. Notes on Interpolation. II . (Explicit formulae). Act a Mathematic a Academia e
Scientiarum Hungaricae , vol . 8 (1 957) , pp . 201 -21 5 .
2. Notes on Interpolation. III. (Convergence). Ibid. , vol . 9 (1 958) , pp . 1 95-21 4 .
BATEMAN, H .
1. A generalization of the Legendre polynomial. Proceeding s o f th e Londo n Mathematica l
Society, ser . 2 , vol . 3 (1 905) , pp . 1 1 1 -1 23 .
2. The solution of linear differential equations by means of definite integrals. Transactions
of th e Cambridg e Philosophica l Society , vol . 2 1 (1 909) , pp . 1 71 -1 96 .
3. Partial Differential Equations. Cambridge Universit y Press , Cambridge , 1 932 .
BATEMAN MANUSCRIP T PROJEC T (Director: A. ERDELYI) .
1. Higher transcendental Functions^ols. 1 , 2, 3 . Ne w York-Toronto-London , 1 953 , 1 955 .
BOCHNER, S. (See List of References above.)
2. Positive zonal functions on spheres. Proceedings o f th e Nationa l Academ y o f Sciences ,
vol. 4 0 (1 954) , pp . 1 1 41 -1 1 47 .
BONAMI, A., AN D CLERC, J.-L.
1. Sommes de Cesaro et multiplicateurs des d&veloppements en harmoniques spteriques.
Transactions o f th e America n Mathematica l Society , vol . 1 8 3 (1 973) , pp . 223-263 .
BUTLEWSKI, Z .
1. Sur les integrates d'une Equation difffrentielle du second ordre. Mathematic a (Cluj) ,
vol. 1 2 (1 936) , pp . 36-48 .
CARLESON, L .
1. On convergence and growth of partial sums of Fourier series. Act a Mathematica ,
vol. 1 1 6 (1 966) , pp . 1 35-1 57 .
COIFMAN, R. , AN D WEISS, G .
1. Analyse harmonique non-commutative sur certaines espaces homogenes. Lectur e Note s
in Mathematics , vol . 242 , Springer-Verlag , Berli n an d Ne w York , 1 971 .
418
FURTHER REFERENCE S
CS0RDAS*G., AN D WILLIAMSON , J .
1. On polynomials satisfying a Turan type inequality. Proceeding s o f th e America n
DAVIS, J. , AN D HIRSCHMAN, 1 .1 . , JR .
1. Toeplitz forms and ultraspherical polynomials.
vol. 1 8 (1 966) , pp . 73-95 .
Pacifi
c Journa l o f Mathematics ,
DAVIS, J. , AN D RABINOWITZ, P .
1. (a ) Some geometrical theorems for abscissas and weights of Gauss type. Journa l o f
Mathematical Analysi s an d Applications , vol . 2 (1 961 ) , pp . 428-437 . (b ) Erratum.
Ibid., vol . 3 (1 961 ) , p . 61 9 .
DE BRUIJN , N . G .
1. Uncertainty principles in Fourier analysis, Inequalities , Edite d b y O . Shisha, Academi c
Press, Ne w York , 1 967 , pp . 57-71 .
DOETSCH, G . (See List of References above. )
3. Handbuch der Laplace-Transformationen. Vol . 2 : Anwendungen der Laplace-Trans formationen, 1 . Abteilung. Basel-Stuttgart , 1 955 .
EAGLESON, G .
1. A characterization theorem for positive definite sequences on the Krawtchouk polynomials.
Australian Journa l o f Statistics , vol . 1 1 (1 969) , pp . 29-38 .
EGERVARY, E. , AN D TURAN, P .
1. Notes on interpolation. V . (On the stability of interpolation). Act
Academiae Scientiaru m Hungaricae , vol . 9 (1 958) , pp . 259-267
2. Notes on interpolation. VI . (On the stability of the interpolation on an
Acta Mathematic a Academia e Scientiaru m Hungaricae , vol . 1 0 (1
ERDELYI, A. (See List of References above. )
3. Asymptotic forms for Laguerre polynomials. Journa l o f th e India
Society, Golde n Jubile e Volume , vol . 2 4 (1 960) , pp . 235-250 .
a Mathematic a
.
infinite interval).
959) , pp . 55-62 .
n Mathematica l
ERDELYI, A. , AN D SWANSON, C . A .
1. Asymptotic forms of Whittaker's confluent hypergeometric functions. Memoir s o f th e
American Mathematica l Society , No . 25 , 1 957 .
ERDOS, P .
1. On the distribution of normal point groups. Proceeding s o f th e Nationa l Academ y o f
Sciences, vol . 2 6 (1 940) , pp . 294-297 .
2. On divergence properties of the Lagrange interpolation parabolas. Annal s o f Mathe matics, vol . 4 2 (1 941 ) , pp . 309-31 5 .
3. Problems and results on the theory of interpolation. II . Act a Mathematic a Academia e
Scientiarum Hungaricae , vol . 1 2 (1 961 ) , pp . 235-244 .
ERDOS, P. , AN D GRUNWALD, G .
1. Ober einen Faber'schen Satz. Annal s o f Mathematics , vol . 3 9 (1 938) , pp . 257-261 .
ERDOS P. , AN D LENGYEL, B . A .
1. On fundamental functions of Lagrangean interpolation. Bulleti n o f th e America n
ERDOS, P. , AND TURAN , P . (See List o f References above. )
3. On Interpolation. III . Interpolator^ theory of polynomials. Annal s o f Mathematics ,
vol. 4 1 (1 940) , pp . 51 0-553 .
4. An extremal problem in the theory of interpolation. Act a Mathematic a Academia e
Scientiarum Hungaricae , vol . 1 2 (1 961 ) , pp . 221 -234 .
FELDHEIM E.(Se e Lis t of References above. )
5. On the positivity of certain sums of ultraspherical polynomials. (Translate d an d edite d
by G . Szego. ) Journa l d'Analys e Mathematique , vol* . 1 1 (1 963) , pp . 275-284 .
FREUD, G .
1. Az Hermite-Fej6r-feIe interpolacids eljards konvergenciajardl. Magya r Tudomanyo s
Akademia l i t . osztalyana k kozlemenyei , vol . 5 (1 955) , pp . 29-47 .
2. Ortogonalis polindmokrdl. Dissertation . Budapest , 1 956 .
3. Ober die Asymptotik orthogonaler Polynome. Academi e Serb e de s Sciences , vol . 1 1
(1957), pp . 1 9-32 .
4. Orthogonale Polynome. Basel , 1 969 . Englis h translation , Ne w York , 1 971 .
FURTHER REFERENCE S
419
GASPER, G .
1. Linearization of the product of Jacobi polynomials. I . Canadia n Journa l o f Mathe matics, vol . 2 2 (1 970) , pp . 1 71 -1 75 .
2. Linearization of the product of Jacobi polynomials. II . Canadia n Journa l o f Mathe matics, vol . 2 2 (1 970) , pp . 582-593 .
3. Positivity and the convolution structure for Jacobi series. Annal s o f Mathematics ,
(2), vol . 9 3 (1 971 ) , pp . 1 1 2-1 1 8 .
4. Banach algebras for Jacobi series and positivity of a kernel. Annals o f Mathematics ,
(2), vol . 9 5 (1 972) , pp . 261 -280 .
5. An inequality of Turan type for Jacobi polynomials. Proceeding s o f th e America n
6. Nonnegativity of a discrete Poisson kernel for the Hahn polynomials. Journa l o f
Mathematical Analysi s an d Applications , vol . 4 2 (1 973) , pp . 438-451 .
7. Projection formulas for orthogonal polynomials of a discrete variable. Journa l o f
Mathematical Analysi s an d Applications , vol . 4 5 (1 974) , pp . 1 76-1 98 .
GATTESCHI, L .
1. Approssimazione asintotica degli zeri dei polinomi ultrasferici. Rendicont i d i Matematic a
e dell e su e Applicazioni , (5) , vol . 8 (1 949) , pp . 399-41 1 .
2. Limitazione degli errori nelle formule asintotiche per le funzioni speciali. Rendicont i
del Seminari o Matematic o delTUniversit a e de l Politecnic o d i Torino , vol . 1 6
(1956-1957), pp . 83-94.
GlNIBRE, R .
1. General formulation of Griffiths' inequalities. Communication s o n Mathematica l
Physics, vol . 1 6 (1 970) , pp . 31 0-328 .
GRENANDER, U., AN D SZEGO, G .
1. Toeplitz Forms and their Applications. Berkeley-Lo s Anageles , 1 958 .
GRONBAUM, F . A .
1. A property of Legendre polynomials. Proceedings of the National Academy o f Sciences ,
vol. 6 7 (1 970) , pp . 959-960 .
2. A new kind of inequality for Bessel functions. Journal o f Mathematica l Analysi s an d
Applications, vol . 4 1 (1 973) , pp . 1 1 5-1 21 .
GRCNWALD, G. (See List of References above.)
2. On a convergence theorem for the Lagrange interpolation polynomials. Bulleti n o f th e
American Mathematica l Society , vol . 4 7 (1 941 ) , pp . 271 -275 .
3. On the theory of Interpolation. Acta Mathematica , vol . 7 5 (1 943) , pp . 21 9-245 .
HAHN, W. (See List of References above.)
5. Vber Orthogonalpolynome, die q-Differenzengleichungen genugen. Mathematisch e
Nachrichten, vol . 2 (1 949) , pp . 4-34 .
HARTMAN, P. , AN D WINTNER, A .
1. On non-conservative linear oscillators of low frequency. American Journa l o f Mathe matics, vol . 7 0 (1 948) , pp . 529-539 .
HIRSCHMAN, 1 .1 . , JR .
1. Variation diminishing transformations and orthogonal polynomials. Journa l d'Analys e
Mathematique, vol . 9 (1 961 ) , pp . 1 77-1 93 .
2. The strong Szegd limit theorem for Toeplitz determinants. America n Journa l o f
Mathematics, vol . 8 8 (1 966) , pp . 73-95 .
HORTON, R .
1. Expansions using orthogonal polynomials. Ph.D . Thesis , Universit y o f Wisconsin ,
Madison, 1 973 , 60 pp.
Hstr, H.
1. Certain integrals and infinite series involving ultraspherical polynomials and Bessel
functions. Duke Mathematica l Journal , vol . 4 (1 938) , pp . 374-383 .
HUA, L . K.
1. Harmonic Analysis of Functions of Several Complex Variables in the Classical Domains,
Translations o f Mathematica l Monographs , vol . 6 (1 963) , America n Mathematica l
Society, Providence , R . I .
420
FURTHER REFERENCE S
HUNT, R
1. On the convergence of Fourier series, Orthogona l Expansion s an d thei r Continuou s
Analogues, Edite d b y D . Haimo , Souther n Illinoi s Universit y Press , Carbondale ,
Illinois, 1 968 , pp . 235-255 .
KARLIN, S. , AN D MCGREGOR , J. L .
1. The differential equations of birth-and-death processes, and the Stieltjes moment problem.
Transactions o f th e America n Mathematica l Society , vol . 8 5 (1 957) , pp . 489-546 .
2. The Hahn polynomials, formulas and an application. Script a Mathematica , vol . 2 6
(1961), pp . 33-46 .
3. Classical diffusion processes and total positivity. Journa l o f Mathematica l Analysi s
and Applications , vol . 1 (1 960) , pp . 1 63-1 83 .
KARLIN, S. , AN D STUDDEN, W . J .
1. Tchebycheff Systems with Applications in Analysis and Statistics. Interscience Publishers,
New York , 1 966 .
KARLIN, S. , AN D SZEGO, G .
1. On certain determinants whose elements are orthogonal polynomials. Journa l d'Analys e
Mathematique, vol . 8 (1 960) , pp . 1 -1 57 .
KOORNWINDER, T .
1. The addition formula for Jacobi polynomials. I . Summary of results. Indagatione s
Mathematicae, vol . 3 4 (1 972) , pp . 1 88-1 91 .
2. The addition formula for Jacobi polynomials and spherical harmonics. SIAM Journa l
on Applie d Mathematics , vol . 2 5 (1 973) , pp . 236-246 .
3. Jacobi polynomials. II. An analytic proof of the product formula. SIAM Journa l o n
Mathematical Analysis , vol . 5 (1 974) , pp . 1 25-1 37 .
4. Jacobi polynomials. III . An analytic proof of the addition formula. SIA M Journa l
on Mathematica l Analysis , vol . 6 (1 975) , pp . 533-543 .
LOCHER, F .
1. Norm bounds of quadrature processes. SIAM Journa l o n Numerica l Analysis , vol .
10 (1 973) , pp . 553-558 .
LORCH, L .
1. The Lebesgue constants for Jacobi series. I. Proceedings o f th e America n Mathematica l
Society, vol . 1 0 (1 959) , pp . 756-761 .
2. The Lebesgue constants for Jacobi series. II . America n Journa l o f Mathematics , vol .
81 (1 959) , pp . 875-888 .
3. Comparison of two formulations of Sonin's theorem and of their respective applications
to Bessel functions. Studi a Scientiaru m Mathematicaru m Hungarica , vol . 1
(1966), pp . 1 41 -1 45 .
LORCH, L. , MULDOON , M . E. , AN D SZEGO, P .
1. Higher monotonicity properties of
Journal o f Mathematics , vol . 2
2. Higher monotonicity properties of
Journal o f Mathematics , vol . 2
certain Sturm-Liouville functions. III . Canadia n
2 (1 970) , pp . 1 238-1 265 .
certain Sturm-Liouville functions. IV . Canadia n
4 (1 972) , pp . 349-368 .
LORCH, L. , AN D SZEGO, P .
1. Higher monotonicity properties of certain Sturm-Liouville functions. Acta Mathematica ,
vol. 1 0 9 (1 963) , pp . 55-73 .
2. Higher monotonicity properties of certain Sturm-Liouville functions. II . Bulleti n d e
l'Academie Polonais e de s Sciences . S6ri e de s Science s Math6matiques , Astro nomiques e t Physiques , vol . 1 1 (1 963) , pp . 455-457 .
MAGNUS, W. , AN D OBERHETTINGER , F .
1. Formeln und Lehrsatze fur die speziellen Funktionen der mathematischen Physik.
Berlin, 1 948 .
MAKAI, E .
1. Ober die Nullstellen von Funktionen, die Lbsungen Sturm-Liouville'scher Differentialgleichungen sind. Commentari i Mathematic i Helvetici , vol . 1 6 (1 944) , pp . 1 53-1 99 .
2. On a monotonic property of certain Sturm-Liouville functions. Act a Mathematic a
Academiae Scientiaru m Hungaricae , vol . 3 (1 952) , pp . 1 65-1 72 .
FURTHER REFERENCE S
421
3. On systems of polynomials orthogonal in two intervals. Publicatione s Mathematicae ,
vol. 2 (1 952) , pp . 222-228 .
MAKAI, E. , AN D TURAN, P .
1. Hermite expansions and distribution of zeros of polynomials. Magya r Tud . Akademi a
Mat. Kutatolnt . Kozl . vol . 8 (1 963) , pp . 1 57-1 63 .
MUCKENHOUPT, B .
1. Asymptotic forms for Laguerre polynomials. Proceedings o f the American Mathematica l
Society, vol . 2 4 (1 970) , pp . 288^292 .
2. Mean convergence of Hermite and Laguerre series. I . Transaction s o f th e America n
Mathematical Society , vol . 1 4 7 (1 970) , pp . 41 9-432 .
3. Mean convergence of Hermite and Laguerre series. II . Transaction s o f th e America n
Mathematical Society , vol . 1 4 7 (1 970) , pp . 433-460 .
4. Equiconvergence and almost everywhere convergence of Hermite? and Laguerre series.
SIAM Journa l o n Mathematica l Analysis , vol . 1 (1 970) , pp . 295-321 .
5. Mean convergence of Jacobi series. Proceeding s o f th e America n Mathematica l
Society, vol . 2 3 (1 969) , pp . 306-31 0 .
MUCKENHOUPT, B. , AN D STEIN, E .
1. Classical expansions and their relation to conjugate harmonic functions. Transaction s
of th e America n Mathematica l Society , vol . 1 1 8 (1 965) , pp . 1 7-92 .
NEWMAN, J. , AN D RUDIN , W .
1. Mean convergence of orthogonal series. Proceeding s o f th e America n Mathematica l
Society, vol . 3 (1 952) , pp . 21 9-222 .
NOVIKOFF, A .
I. On a special system of orthogonal polynomials. Dissertation, Stanfor d University , 1 954 .
OLVER, F . W . J .
1. A paradox in asymptotics. SIA M Journa l o n Mathematica l Analysis , vol . 1 (1 970) ,
pp. 533-534 .
PEETRE, J .
1. The Weyl transform and Laguerre polynomials, L e Matematiche , vol . 2 7 (1 972) ,
pp. 301 -323 .
POLLACZEK, F .
1. Sur une generalisation des polynomes de Legendre. Comptes Rendu s d e l'Academi e
des Sciences , Paris , vol . 22 8 (1 949) , pp . 1 363-1 365 .
2. Syst&mes de polynomes biorthogonaux qui generalised les polynomes ultraspheriques.
Ibid., vol . 22 8 (1 949) , pp . 1 998-2000 .
3. Sur une famille de polynomes orthogonaux qui contient les polynomes d'Hermite et de
Laguerre comme cas limites. Ibid. , vol . 23 0 (1 950) , pp . 1 563-1 565 .
4. Sur une generalisation des polynomes de Jacobi. Memorial de s Sciences Mathematiques ,
vol. 1 3 1 (1 956) .
POLLARD, H .
1. The mean convergence of orthogonal series. I. Transactions o f the American Mathematica l
Society, vol . 6 2 (1 947) , pp . 387-403 .
2. The mean convergence of orthogonal series. II . Transaction s o f th e America n Mathe matical Society, vol . 6 3 (1 948) , pp . 355-367 .
3. The mean convergence of orthogonal series. III . Duk e Mathematica l Journal , vol .
16 (1 949) , pp . 1 89-1 91 .
ROOSENRAAD, C . T .
1. Inequalities with orthogonal polynomials. Ph.D . Thesis , Universit y o f Wisconsin ,
Madison, 1 969 , 5 2 pp .
SAPIRO, R . L .
1. The special functions connected with the representations of the group SU(n) of class
one with respect to SU(n — 1 ) (n ^ 3) . Izvestij a Vyssi h Ucebny h Zavedenii ,
Matematika, vol . 7 1 (1 968) , pp . 97-1 07 , (Russian) .
SARMANOV, I . O .
1. A generalized symmetric gamma correlation. (Russian; translatio n in ) Sovie t Mathe matics, Doklady , vol . 9 (1 968) , pp . 547-550 .
422
FURTHER REFERENCE S
SCHMEISSER, G .
1. Optimale Schranken zu einem Satz uber Nullstellen Hermitescher Trinome. Journa l
fur di e rein e un d angewandt e Mathematik , vol . 24 6 (1 971 ) , pp . 1 47-1 60 .
SCHOENBERG, I. , AN D SZEGO, G .
1. An extremum problem for polynomials. Compositi o Mathematica , vol .1
pp. 260-268 .
4 (1 960) ,
SEIDEL, W. , AN D SZASZ, 0 .
1. On positive harmonic functions and ultraspherical polynomials. Journal o f th e Londo n
Mathematical Society , vol . 2 6 (1 951 ) , pp . 36-41 .
SHOHAT, J . A. , HILLE , E. , AN D WALSH , J . L .
1. A Bibliography on Orthogonal Polynomials. Washington, 1 940 .
SURANYI, J. , AN D TURAN, P .
1. Notes on interpolation. I . (On some interpolatorical properties of the ultraspherical
polynomials). Act a Mathematic a Academia e Scientiaru m Hungaricae , vol . 6
(1955), pp . 67-79 .
SzAsz, O. (See List of References above.)
2. On the relative extrema of ultraspherical polynomials. Bollettino dell a Unione Matematic a
Italiana, serie s 7 , vol . 5 (1 950) , pp . 1 25-1 27 .
3. On the relative extrema of the Hermite orthogonal functions. Journa l o f th e India n
Mathematical Society , vol . 2 5 (1 951 ) , pp . 1 29-1 34 .
SZEGO, G. (See List of References above.)
22. On an inequality of P. Turan concerning Legendre polynomials. Bulleti n o f th e
American Mathematica l Society , vol . 5 4 (1 948) , pp . 401 -405 .
23. On the relative extrema of Legendre polynomials. Bollettin o dell a Union e Matematic a
Italiana, (3) , vol . 5 (1 950) , pp . 1 20-1 21 .
24. On certain special sets of orthogonal polynomials. Proceeding s o f th e America n
Mathematical Society , vol . 1 (1 950) , pp . 731 -737 .
25. Ultrasphaerikus polinomok dsszegtrol. Matematika i e s Fizika i Lapok , vol . 4 5
(1938), pp . 36-38 .
26. Uber gewisse Potenzreihen mit lauter positiven Koeffizienten. Mathematisch e
Zeitschrift, vol . 3 7 (1 933) , pp . 674-688 .
27. On some Hermitian forms associated with two given curves of the complex plane.
Transactions o f th e America n Mathematica l Society , vol . 4 0 (1 936) , pp . 450-461 .
SZEGO, G. , AN D TURAN, P .
1. On the monotone convergence of certain Riemann sums. Publicatione s Mathematica e
(Debrecen), vol . 8 (1 961 ) , pp . 326-335 .
THORNE, R . C .
1. The asymptotic expansion of Legendre functions of large degree and order. Technica l
Report, Offic e o f Nava l Research , Californi a Institut e o f Technology , 1 956 .
TRICOMI, F. G. (See List of References above.)
2. Sul comportamento asintotico dell'n-esimo polinomio di Laguerre nelVintorno dell'ascissa
4n. Commentari i Mathematic i Helvetici , vol . 2 2 (1 949) , pp . 1 50-1 67 .
3. Sul comportamento asintotico dei polinomi di Laguerre. Annali d i Matematica , (4) ,
vol. 2 8 (1 949) , pp . 263-289 .
4. Sugli zeri dei polinomi sferici ed ultrasferici. Ibid. , (4) , vol . 3 1 (1 950) , pp . 93-97 .
5. Vorlesungen uber Orthogonalreihen. Berlin-Gottingen-Heidelberg , 1 955 .
TURAN, P .
1. On the zeros of the polynomials of Legendre. Casopis pr o Pestovan i Matematik y a
Fysiky, vol . 7 5 (1 950) , pp . 1 1 3-1 22 .
2. On Descartes-Herriot's rule. Bulleti n o f th e America n Mathematica l Society , vol .
55 (1 949) , pp . 797-800 .
3. Remark on a theorem of Erhard Schmidt. Mathematica , vol . 2 (1 960) , pp . 373-378 .
VIETORIS, L .
1. Uber das Vorzeichen gewisser trigonometrischer Summen. Sitzungsbericht e de r
mathematisch-naturwissenschaftlichen Klass e de r Akademi e de r Wissenschafte n
in Wien , vol . 1 6 7 (1 958) , pp . 1 25-1 35 .
FURTHER REFERENCE S
423
VlTALI, G. , AN D SANSONE, G .
See Lis t o f Reference s above . Thir d edition . Bologna , 1 952 .
WEBSTER, M . S .
1. A convergence theorem for certain Lagrange interpolation polynomials. Bulleti n o f th e
American Mathematica l Society , vol . 4 9 (1 943) , pp . 1 1 4-1 1 9 .
WlDOM, H . , AN D WlLF, H .
1. Small eigenvalues of large Hankel matrices. Proceedings o f th e America n Mathematica l
Society, vol . 1 7 (1 966) , pp . 338-344 .
ZYGMUND, A .
See Lis t o f Reference s above . Secon d edition . Ne w York , 1 952 .
INDEX
The numbers refer to pages
of Mehler-Heine type, 193,194,
Abel, v, 100,401.
— *s continuity theorem, 92.
— 's transformation, 2,91,135.
— summability, 245,273.
See Inequality.
Achieser, 5, 37, 42,401,416.
Adamoff, 203,248,249,250,401 .
Addition theorems, 58,97 ff.
Additive operation, 11,12.
Airy's function , 1 8 , 1 9 , 1 31 , 201 , 234 , 239 ,
243, 377,382.
See Zeros.
Antipole condition, 246, 249, 265.
Approximation, 5 ff.
Askey, xi , 98 , 99 , 1 1 0 , 273 , 348 , 363 , 391 ,
392,416.
Askey-Fitch, 96,363,416.
Askey-Gasper, 97,99,1 1 0,1 90,392,41 7 .
Askey-Pollard, 273,417.
Askey-Steinig, 97,158,363,417.
Askey-Wainger, 99,190, 273,417.
Associated function, Legendre, 84.
Asymptotic formul a o f Besse l functions ,
15,16.
classical polynomials, 191 ff.
— — — genera l orthogona l polynomials , v ,
vi, 296 ff.
— — — Hermite polynomials, 1 32 , 1 91 , 194,
198 ff., 218 ff., 235,242.
— — — Jacobi function s o f secon d kind ,
201, 225.
polynomials, 58, 1 67 , 1 68 , 1 9 1 ff.,
201, 202,212,225.
0 f HuV s type , 1 69 , 1 97 , 202 ,
214.
0 f Mehler-Hein e type ,1 67 ,
192 ff.,202.
See Darboux's formula.
kernel polynomials, 369 ff.
Laguerre polynomials, 1 32 , 1 91 , 193,
194,198 ff., 236, 237, 240 ff.
of Hilb' s type , 1 77 , 1 99 , 203 ,
216 ff.,219 ff.,384.
217.
ofPlancherel-Rotach type , 200,
201,227 ff.
SeeFejer's formula , Perron's formula.
Legendre function s o f secon d kind ,
198,212, 222 ff.
polynomials, 191 , 193, 194 ff., 243.
Asymptotic formula of Legendre polynomial s
of Hub's type, 195,202,212 ff.
See Laplac e formula , Laplace-Hein e
formula.
polynomials orthogona l o n a curve ,
371 ff.
the unit circle, 297 ff.
ultraspherical polynomials , 1 96 , 1 97 ,
206, 208, 209, 212.
Bailey, 273,392,417.
Balazs-Turan, 347,417.
Banach, 13,401.
Bateman, 96, 98, 99, 273, 417.
Bateman Project, ix , 95 , 1 1 0 , 243 , 393 , 394,
417.
Bernstein, S., vi , 9, 31 , 42, 1 59 , 1 65 , 1 6 8 ff.,
296, 299,300, 303, 314,315, 330, 401.
— 's theore m o n trigonometri c polynomials ,
5, 280, 304.
See Polynomials.
Bessel functions , v , 1 4 ff., 95 , 96 , 1 0 2 ff.,
126 ff., 1 40 , 1 66 , 1 67 , 1 91 , 1 93 , 202 , 203 ,
212 ff., 225 , 243 , 272 , 353 , 362 , 363 , 380 ,
384, 387,392, 400.
See Asymptotic formula, Zeros.
- ' s inequality, 25, 38, 289, 316, 322, 333, 367.
Blumenthal, 268,310, 401.
Bochner, 108, 273, 401,417.
Bonami-Clerc, 273,417.
Bottema, 132,401.
Bounded variation, functions of, 12.
Brauer, A., 131,401.
Bruns, 122,125,136,138, 401.
425
426
INDEX
Buell, 124 ff.,401.
Butlewski, 166, 417.
Capacity, see Transfinite diameter.
Caratheodory, 364, 402.
See Osgood-Caratheodory.
Carleman, 366, 402.
Carleson, 273, 417.
Cauchy-Hadamard formula, 248, 253, 312.
Cauchy's principle value, 278.
-theorem, 69, 1 06 , 221 , 275 , 289 , 324 , 370 ,
372.
Centroid, 189.
Cesaro summability, vi, 244, 246 ff., 258.
(means) o f Fourie r series , 1 2 , 1 4 , 246 ,
250.
Hermite series, 250, 251 .
Jacobi series, vi, 246, 248, 249, 256 ff.
Cesaro summabilit y o f Laguerr e series , 247 ,
248, 250, 251,271 ff.
Legendre series, 14, 249.
power series, 325.
— ultraspherical series, 248.
Characteristic values, 49,187.
Charlier, see Polynomials.
Christoffel, iii, 42, 43, 47, 402.
- (-Darboux ) formula, 42 ff., 318, 326.
- formul a of, 29 ff., 397.
-numbers, 4 6 ff., 114, 1 1 5 , 187, 351 ff., 378.
Clerc, see Bonomi-Clerc.
Closure, see Orthogonal polynomials.
Coifman-G. Weiss, 273, 417.
Completely monotonic sequence, 1 3 6 ff., 155.
Confluent hypergeometric function, 104.
Conformal mapping , 21 , 1 60 , 364 , 365 , 372 .
Conjugate function, 279.
- point s (interpolation), 332, 347.
Continued fractions, v, 54 ff.
Continuous operation, 12.
Convergence in mean, 330.
, generalized, 330.
- o f interpolation, 330.
- , quadrature , 330, 333,350 ff.
Convergent (continued fraction), 55 ff.
Cotes numbers, 12, 349 ff.,363.
Courant-Hilbert, see Hilbert-Courant.
Cramer, 251, 402.
Csordas-Williamson, 388, 418.
Darboux, 42, 43,195,196, 211, 402.
Darboux's formul a fo r Jacob i polynomials ,
168, 196, 225, 236, 237, 248, 253, 337, 353 .
- method , 202, 206 ff., 265, 269, 395.
Davis-Hirschman, 99, 418.
Davis-Rabinowitz, 1 58 , 418.
DeBruijn, 110,418.
de la Vallee-Poisson summability, 273.
Deviation, quadratic, 38, 41, 288.
-,Tchebichef,41,368.
Differences, 34, 35,136.
Differential equation , v , 1 6 ff., 37 , 1 52 , 1 59 ,
164,166, 210.
of Bessel functions, 15.
Hermite polynomials , 1 06 , 1 76 , 380 .
— Jacobi (hypergeometric) polynomials ,
iv,60ff., 1 1 7,1 41 .
Laguerre polynomials, 1 00 , 1 1 7 , 1 76 .
ultraspherical polynomials , 80 , 81 .
Dirichlet, 85, 402.
Dirichlet-Mehler, see Integral representation.
Dirichlet's integral, 12,14.
Discriminant of classical polynomials , 1 4 3 ff.
Distribution, 8, 9, 26, 29, 38, 40, 57, 1 81 , 274,
330, 351,378.
- o f Stieltjes type, 9, 33 ff.,38.
Doetsch, 34, 35, 380, 402, 418.
Du Bois, Reymond, 14, 402.
Eagleson, 37, 418.
Egervary-Turan, 348, 418.
Ellipse, 8, 21, 251 ff.
- o f convergence , 245 , 248 , 252 , 31 1 , 31 2 .
Elliptic functions, v, 60.
Equiconvergence, 31 , 244 , 246 , 247 , 249 ,
313 ff.
Erdelyi, ix, 104, 243, 273, 380, 402, 418.
Erdelyi-Swanson, 243,418.
Erdos, vii, 347, 348, 418.
Erdos-Feldheim, 334, 402.
Erdos-Grunwald, 347,418.
Erdos-Lengyel, 347, 418.
Erdos-Turan, 1 1 3 , 1 1 4 , 1 1 5 , 332 , 333 , 335 ,
347, 348, 402, 418.
Euler, 72,402.
— 's constant, 15.
— 's integral of the first kind, 14.
second kind, see Gamma function .
Expansion, v, vi, 58.
—, finite cosine , o f Legendr e polynomials ,
90 ff.
—, infinit e sine , o f Legendr e polynomials ,
90 ff.
— in power series, 313, 314.
— in serie s o f classica l polynomials , 24 4 ff.,
251 ff.
INDEX
general orthogona l polynomials ,
313 ff.
Hermite polynomials , 244 , 245 ,
247,251, 253, 269 ff.
Jacobi polynomials, 244 ff.
Laguerre polynomials , 24 4 ff.,
249 ff.,253, 266 ff.,313.
Legendre polynomials , 248 , 249 .
Tchebichef polynomials , 248 , 314.
ultraspherical polynomials , 248 ,
249.
See Fourie r series , Orthogona l poly nomials.
427
Fundamental polynomial s o f Hermit e inter polation, 330 ff.
Lagrange interpolation, 12, 47, 329 ff.
Galbrun, 251,404.
Gamma-function, 1 4 , 75.
Gasper, 37, 98, 99,190, 273, 419.
See Askey-Gasper.
Gatteschi, 242, 419.
Gauss, iii, 33,47, 49,62, 404.
See Mean-value theorem , Mechanica l
quadrature.
Gauss-Weierstrass summability, 273.
Gegenbauer, 80,94,98,99,404.
"Gegenbeispiel", 250, 272.
Faber, 299, 330, 368, 369, 372, 402.
Generating function, 35, 36.
See Polynomials.
of Hermite polynomials, 106, 380.
Fatou, 275,403.
Jacobi polynomials, 6 9 ff., 82, 83, 95.
— 's theorem, 274.
Laguerre polynomials, 1 01 , 202 , 379.
Favard, 43, 403.
Legendre polynomials , 90 , 206 , 207 ,
Fejer, vii, 1 4 , 45, 58, 89, 91, 96, 1 34 , 1 3 6 ff.,
383.
157, 165, 172, 174, 1 75 , 179, 198, 202, 249 ,
Pollaczek polynomials, 393.
264, 330 ff., 336, 348, 350, 386,403.
See Stekloff-Feje>.
—'s asymptoti c formul a fo r Laguerr e poly Geometric mean, 275, 296,385.
nomials, 198, 202, 203,237,240,269.
Geronimus, 42,1 89 , 404.
Fejer's generalizatio n o f Legendr e poly Gibbs' phenomenon, 251.
nomials, 135 ff., 174 ff., 206.
Ginibre, 392,419.
— 's second generalizatio n o f Legendr e poly - Gottlieb, 37,405.
nomials, 137,138,155.
Grenander-Szego, 274, 287,419.
Gronwall, 165, 249, 405.
— 's integral, 12.
— *s representation o f positiv e trigonometri c Grlinbaum, 392,419.
Griinwald, vii, 330,347, 405,419.
polynomials, 3,4,274.
See Erdos-Grunwald.
9 generalization of, 275 ff.
Fejer-Szego, 175, 404.
Griinwald-Turan, 346,405.
Fekete, 369,404.
Feldheim, 95, 96, 97, 334, 335, 386, 404, 418.
Haar, 14, 248,405.
See Erdos-Feldheim.
Hahn, W. , 33 , 1 07 , 1 1 1 , 1 1 2 , 1 29 , 1 32 , 151 ,
Fitch, see Askey-Fitch.
405,419.
Fourier, v.
Hamburger, 57,110,405.
— coefficient (constant) , 24,287.
Hankel, see Quadratic form.
— 's inversion formula, 387.
Hardy, 102,380,405.
-series, 1 2 , 14, 24, 25, 28 , 38 , 39 , 244 , 246 , Hardy-Littlewood-Polya, 2,405 .
253, 274, 289, 311, 314, 323, 347, 367.
Hartman-Wintner, 20,419.
See Cesaro means.
Hausdorff, 136,155,405.
Freud, 309, 418.
Heine, iii , 27 , 37 , 47 , 54 , 80 , 9 0 ff., 1 5 1 ff.,
Friedrichs, 404.
192,194, 251, 405.
Fujiwara, 145, 404.
Heine-Stieltjes theorem, iv, 151 ff.
Functions of second kind, Jacobi's, 73 ff., 251. Helly, 13, 405.
, Legendre's , 74 , 78 , 88 , 89 , 92 ,
- ' s theorem , 1 3 , 14, 258, 272, 330, 339, 350 .
379, 383.
Hermite, v, 156,405, 406.
See Asymptotic formula, Integral
- interpolation , 330 ff., 340 ff., 347, 348.
representation, Zeros.
Hermite-Stieltjes, 89,1 56,1 57,1 72,406 .
428
INDEX
Hermitian form , 287.
Hilb, 1 95 , 249,406 .
Hilbert, 1 42,1 45,406 .
Hilbert-Courant, 58 , 59 , 1 00 , 1 06 , 1 09 , 406 .
Hildebrandt, 8,1 0 , 406 .
Hille, 1 02,1 05,1 26,1 31 ,242 , 251 ,406 .
See Shohat-Hille-Walsh .
Hirschman, 99, 273, 419.
See Davis-Hirschman .
Hobson, iv, 58,84,92, 382 , 406.
HoU6, 335,406 .
Horton, 99,273,41 9 .
l'Hospital, rul e of, 307 .
Howell, 392 .
Hsu, 390,41 9 .
Hua, 99,41 9 .
Hunt, 273 , 420.
Hurwitz, theore m of , 21 , 1 50 , 1 93 , 239 , 371 .
Hypergeometric function , iii , 6 2 ff., 83 , 84 ,
96.
See Differential equation .
Inequalities, 1 5 9 ff.
Inequality, Abel's, 2,174, 205 .
- , Cauchy's , 2 , 39 , 1 20 , 1 60 , 1 83 , 291 , 302 ,
306,321,370.
— for the arithmeti c an d geometri c mean , 2 ,
300.
- , Schwarz's , 2 , 9 , 1 1 0 , 1 36 , 1 62 , 268 , 276 ,
317, 376.
- , Turan's , 1 90 , 388.
See Bessel's inequality .
Integral, Lebesgue , vii , 9 , 26 , 1 59 , 24 6 ff.,
274,275,287,291,364,384.
- , Riemann , 9 , 1 0 , 280 , 281 , 282 , 298 , 31 0 ,
333, 335, 351, 361 .
- , Riemann-Stieltjes , 8 , 9 , 1 1 , 50 , 333 , 350 ,
351.
- , Stieltjes-Lebesgue , v , 1 ,8,1 0,38,1 87 .
— equations, v , 218, 251.
— representatio n o f Legendr e function s o f
second kind, 88 ff.
polynomials, 85 ff.
, Dirichlet-Mehler, 8 5 ff., 96.
, Laplace (first) , 86,1 76 .
, Laplace (second) , 86 ff.
, Stieltjes, 87 ff.
ultraspherical polynomials, Dirichlet Mehler, 89.
, Stieltjes, 89 ff.
Interpolation, v , ix , 1 2 , 1 4 , 58 , 32 9 ff., 347 .
- , Lagrange , 1 2 , 47 , 1 80 , 32 9 ff., 348 , 349 ,
389.
— on Hermite abscissas , 340.
Jacobi abscissas, 335 ff.
Laguerre abscissas , 340, 344 ff.
Legendre abscissas, 386.
Tchebichef abscissas , 330, 334 , 335 , 386 .
ultraspherical abscissas , 386.
See Conjugat e points , Fundamenta l
polynomials, Hermit e interpolation ,
Lagrange polynomials .
Jackson, vii, 6, 7,42, 331 , 406.
Jacob,251,406.
Jacobi, iii, 49, 58,69,86,406,407 .
See Function s o f secon d kind , Me chanical quadrature , Polynomials ,
Series.
Jensen's theorem, 301.
Joo, 391.
Jordan, C, 58 , 407.
- arc , 8, 364.
- c u r v e , 8 , 21 ,364,365 .
- , Ch. , 34, 407.
Jordan-Pochhammer integral , 75.
Julia, 369 , 407.
Kaczmarz-Steinhaus, 1 ,1 0 , 407 .
Kanter, vii .
Karlin-McGregor, 37 , 273, 389,420.
Karlin-Studden, 5 , 420.
Karlin-Szego, 1 90 , 388, 420.
Keldysch-Lawrentieff, 368,407 .
Kernel polynomials, 39 , 40 , 44 , 71 , 1 01 , 1 80 ,
183, 249 , 290 , 322 , 323 , 333 , 351 , 368 , 377 .
See Asymptotic formula , Zeros .
Klein, 1 45,407 .
Kogbetliantz, 1 02 , 1 68 , 1 72 , 203 , 240 , 248 ,
249, 251 , 256, 273,380, 407 .
Koornwinder, 98,99,420 .
Korous, 131,132,162,167, 21 2 , 251, 303, 408 .
Koschmieder, 60,408 .
Kowalewski, 24,408 .
Kowallik, 251 ,408 .
Krall, 1 07 , 408.
Krawtchouk, 1 1 3,408 .
See Polynomials.
Kronecker, 408.
Lagrange, 1 00,408 .
- polynomials , 32 9 ff., 347.
— series, 70.
See Interpolation .
Laguerre, v. 1 00,1 1 7,1 31 ,408 , 409 .
See Polynomials .
INDEX
Lame function, 151.
Langer, 204, 210, 409.
Laplace, iv, 86.
Laplace's formul a fo r Legendr e polynomials ,
194,198, 201, 202, 204 ff., 211 ff.,224.
1 Darboux's generalization , 1 95 ,
201, 206 ff.,211
.
, Stieltjes ' generalization , 1 95 f
202,209 ff.
Laplace-Heine formula, 1 94 , 204 ff., 208, 243.
, generalization, 194.
Laplace series, 96, 249.
- transform , 379.
See Integral representation .
Laurent expansion, 252.
Lawton, 151,409.
Lebesgue, 14,15,409.
See Integral.
- constant , 1 3 , 258, 330, 336 , 338 , 339 , 350 .
Legendre, v, 70, 409.
See Associated function , Polynomials .
Lengyel, 113.
See Erdos-Lengyel.
LeRoy, 103,409.
Level curve, 8.
Leibniz, rule of, 67,68,101.
Limited operation, 12.
Linear operation, see Operation.
Liouville-Stekloff, metho d of , 20 2 ff., 21 0 ff.,
299.
Lipschitz, 383.
- condition , 6,1 62,1 63,1 86 .
Lipschitz-Dini condition, 279, 297, 324.
Littlewood, see Hardy-Littlewood-Polya.
Locher, 363, 420.
Lorch, xi, 190, 249, 420.
Lorch-Muldoon-P. Szego, 158, 420.
Lorch-P. Szego, 158, 420.
Lukacs, 4,178 ff., 249, 409.
Magnus-Oberhettinger, 420.
Makai, 20,148,157,190, 420, 421.
Makai-Turan, 1 58 , 421.
Marcinkiewicz, 273, 330, 347, 409.
Markoff, A. , v , 33, 37, 50, 57, 1 1 5 , 1 1 6 , 121 ,
122,139, 259, 378, 409.
McGregor, see Karlin-McGregor.
Mean approximation, 1 0,1 1 .
Mean-Value theorem, 383.
of Gauss, 275, 310,311.
, second, 2, 202, 357, 363.
Mechanical quadrature, v, ix, 1 2 , 14, 58, 187,
329,348 ff.
429
for classical abscissas, 352 ff.
Jacobi abscissas, 355 ff., 378, 379.
, Gauss-Jacobi, 47 ff., I l l, 34 8 ff.
Mehler, iii, 47, 49,85,192, 380 , 394, 409, 410.
Mehler-Heine, see Asymptotic formula .
Meixner, 34, 35, 410.
Method o f Liouville-Stekloff , se e Liouville Stekloff.
steepest descent, 202,203, 221 ff.
Modulus o f continuity , 6 , 7 , 335 , 336 , 340 ,
346.
Moecklin, 194, 203, 410.
Moment problem of Stieltjes, iii, iv, v, 40.
Muckenhoupt, 1 90 , 243, 273, 421.
Muckenhoupt-Stein, 273,421 .
Muldoon, see Lorch-Muldoon-P. Szego.
Mtintz, 251, 410.
Myller-Lebedeff, 251,-410.
Neumann, E. R., 1 30 , 251, 380,410.
Neumann, F., 248.
Neumann, J. von, 108.
Newman-Rudin, 273, 421.
Newton's formula, 259.
Normalization, 28, 58,160.
Norm function, se e Weight function .
Norm of operation, 12.
Novikoff, 394, 396, 421.
Obrechkoff, 1 98 , 248, 249, 410.
Olver, 243, 421.
Operation, linear functional, 1 1 ff.
Orthogonal polynomials and continued
fractions, 5 4 ff.
— — , asymptotic formul a of , v , ix , 4, 296 ff.
, Christoffel-Darboux formul a for , 4 2 ff.
, classical, 29, 49.
, closure, 40,108 ff.
, definition, ix, 23 ff.
, expansion in serie s of, v , ix , 28, 38, 39,
41, 311, 312, 313 ff.
, extremum properties of, 28, 38 ff.
, general properties of, 38 ff.
, highest coefficient of , 28.
, recurrence formul a for , 4 2 ff., 55 , 391.
, representation of , 27.
, zero s of, v , ix , 44 ff., 121 ff., 1 88 , 1 89 .
Orthogonality, Orthogonalization , v , vi , 8 ,
23 ff.,44.
Orthonormal set, 23, 25, 68.
Osgood-Caratheodory's theorem, 364.
Parabola of convergence, 253.
430
INDEX
Parseval's formula, 40, 289, 368.
Peano, 1.
Peetre, 110, 421.
Pencil, 49.
Perron, 45, 54,144, 202, 410.
—'s formula fo r Laguerr e polynomials , 1 98 ,
199, 202, 203, 220, 221 , 225 ff.
P-function, contiguous Riemann, 71.
Plancherel, 250.
Plancherel-Rotach, 132,201 , 203,410.
Pochhammer-Barnes, notation of, 103.
Poincare, 310.
Poisson's integral, 276, 292.
See Polynomials.
Pollaczek, ix, 37, 393, 421.
Pollard, 273,421.
See Askey-Pollard.
Polya, vii , 42 , 53 , 1 1 7 , 1 34 , 1 54 , 1 66 , 350 ,
388,410,411.
See Hardy-Littlewood-Polya.
P61ya-Szego, vii, 5, 1 1 , 21, 24, 34, 40, 70, 87,
105, 117, 1 34 , 135, 175, 178, 179, 1 84 , 212,
310, 311, 411.
Polynomials associated with a curve
(Tchebichef polynomials), 368, 385.
- , classical , v, 29,159.
See Asymptoti c formula , Expansion ,
Mechanical quadrature.
-,Faber,372,374.
—, Fejer' s generalizatio n o f Legendre , se e
Fejer.
— , Gegenbauer, see Polynomials, ultraspherical.
- , Hermite , vi , ix , 29 , 36 , 37 , 1 0 5 ff., 1 1 0 ,
111, 176 ff.,190, 331, 388 ff.,392.
See Asymptoti c formula , Differentia l
equation, Expansion , Generatin g
function, Interpolation , Recurrenc e
formula, Rodriques* formula, Zeros.
- , Jacobi , vi, ix, 3, 29, 58 ff., 94 ff., 103, 105,
107, 161 , 167 ff., 1 7 2 ff., 1 7 9 ff., 243 , 249 ,
295, 348, 383, 385, 399.
function, Interpolation , Mechanica l
quadrature, Recurrenc e formula ,
Rodrigues' formula, Zeros.
- , Laguerre , vi , ix, 29, 35, 100 ff., 1 1 0 , 111 ,
164, 1 7 6 ff., 1 84 , 1 85 , 1 90 , 243 , 379 , 380 ,
382, 387 ff.,391,393, 395.
function, Interpolation , Recurrenc e
formula, Rodrigues ' formula .
- , Legendre , vi, 29, 30, 33, 34, 48, 58, 63, 70,
85 ff., 95, 96, 1 3 6 ff., 162, 164 ff., 167, 172,
189, 346, 348, 379, 382 ff.,392, 393.
See Asymptoti c formula , Expansion ,
Generating function, Integral representation, Interpolation, Zeros.
—, Krawtchouk, 35 ff.
—, of S. Bernstein and Szego, 31 ff.
- , Poisson-Charlier , 34, 35,37, 377, 389, 390.
- , Pollaczek , 37,393 ff.
— orthogonal on a curve, vii, 364 ff.
See Asymptotic formula, Zeros.
segment, se e Orthogona l poly nomials.
Polynomials orthogonal on the unit circle , ix,
287 ff.,384.
- , Stieltjes-Wigert , 33.
—, Tchebichef (of the first and second kind),
3, 26, 29, 30, 60, 63, 112, 136 ff., 1 62 , 347,
348, 387, 391.
See Expansion , Interpolation , Zeros .
- , ultraspherical , vi , 29 , 58 ff., 80 ff., 93 ff.,
107,135 ff., 167, 170 ff., 379, 387, 390, 394.
function, Integra l representation ,
Interpolation, Recurrenc e formula ,
Rodrigues' formula, Zeros.
Popoviciu, 46,140,142, 411.
Principle of argument, 21,157.
Probability, calculus of, 35.
Quadratic form, 24,123,187, 366.
of Hanke l (recurrent ) type , 27 , 309 .
Quantum mechanics, iii.
Rabinowitz, see Davis-Rabinowitz.
Rau, 197, 214, 249, 411.
Recurrence formula, general, 42 ff.
of Hermite polynomials, 106.
Jacobi functions of second kind, 78 ff.,
379.
polynomials, 71 ff.
Laguerre polynomials, 101.
polynomials orthogona l o n th e uni t
circle, 293.
Riemann, see Integral, P-function .
— 's lemma, 254, 267, 319.
— 's theory of trigonometric series, 248.
Riesz, F„ 12, 275,411 .
Riesz, M., 5, 57,304, 411.
Robin's constant , se e Transfinit e diameter .
INDEX
Rodrigues' formula for Hermite polynomials,
106.
Jacobi polynomials, 67 ff., 73, 74, 94,
99,117.
Laguerre polynomials, 1 01 , 1 1 7 , 388 .
ultraspherical polynomials, 81.
Roever, vii.
Rogosinski, 347.
Rolle's theorem, 51, 53,117, 378.
Roosenrad, 110, 421 .
Ross, vii.
Rotach, 203, 249, 250, 411.
See Plancherel-Rotach.
Rouche's theorem, 21,149.
Rudin, see Newman-Rudin.
Runge, 7.
Runge-Walsh, theorem of, 7.
Saddle-point, 219, 231.
Sapiro, 98, 421.
Sarmonov, 392,421.
Scalar product, 8,9,10, 25, 365, 367, 378.
Schmeiser, 158,422.
Schmidt, E., 34, 35, 411.
Schoenberg, 273.
Schoenberg-Szego, 190, 422.
Schur, I., vi, 140,142, 411.
Schwid, 204,416.
Seidel-Szasz, 94,422.
Sen-Rangachariar, 145,411.
Series, Jacobi, vi, 273.
—, Laguerre, 240.
—, Legendre, vi, 42.
See Expansion, Orthogonal polynomials.
Sherman, 57, 411.
Shibata, 145, 412.
Shohat, v, vii, 42, 48, 159, 163, 189, 309, 333,
336, 340,345,347, 350,385, 386,412.
Shohat-Hille-Walsh, 422.
Singular integral, 13.
Smirnoff, 275, 276, 289, 368, 369, 412.
Smith, 412.
Sonin, 96, 1 00 , 1 02 , 1 04 , 1 59 , 1 66 , 1 69 , 1 76 ,
189, 379,412.
Spencer, 132,412.
Statistics, mathematical, v.
Stein, see Muckenhoupt-Stein.
Steinig, see Askey-Steinig.
Stekloff, 9, 210, 350, 412.
See Liouville-Stekloff.
Stekloff-Fejer, theore m of, 350 ff., 360.
Step function, 34, 35, 37,164.
431
- polynomial , 329, 339 ff.
, generalized, 331.
Stieltjes, v, 33 , 37, 46, 50 , 51 , 53, 54 , 58 , 87 ,
89, 90, 92 , 93 , 1 2 1 ff., 1 36 , 1 39 , 1 40 , 1 42 ,
145, 1 5 1 ff., 1 56 , 1 65 , 1 72 , 1 74 , 1 75 , 1 93 ,
195, 211, 382, 412, 413.
See Heine-Stieltjes , Hermite-Stieltjes ,
Integral, Laplace , Moments , Poly nomials.
Stirling's formula, 227.
— series, 212.
Stone, 10, 23, 251, 413.
Strip of convergence, 253.
Studden, see Karlin-Studden.
Sturm-Liouville type, 210.
Sturm's theore m o n differentia l equations ,
19 ff.
(method) — zeros, vi, 45, 111, 121 ,
124 ff.,139.
Summability, see Abel, Cesaro.
Suranyi-Turan, 347, 422.
Szasz, 5,190, 413, 422.
See Seidel-Szasz.
Szego, G., 19, 27,31, 33, 37,45, 59, 87, 93, 94,
96,110, 1 25 , 1 27 , 1 28 , 1 35 , 1 37 , 1 40 , 1 58 ,
163, 1 64 , 1 6 7 ff., 1 89 , 1 90 , 1 97 , 1 98 , 203 ,
206, 21 4 , 243 , 248 , 249 , 251 , 27 4 ff., 287 ,
289, 29 4 ff., 299 , 300 , 309 , 31 0 , 31 2 , 31 3 ,
340, 346, 355, 360, 361 , 366, 369, 371 , 377,
384, 388,392, 393, 394, 413, 414, 422.
See Fejer-Szego , Grenander-Szego ,
Karlin-Szego, Polya-Szego , Poly nomials, Schoenberg-Szego.
Szego, P., xi.
See Lorch-P . Szego , Lorch-Muldoon P. Szego.
Szego-Turan, 157,158, 422.
Tamarkin, vii, ix, 42, 414.
Tchakaloff, 189, 383, 414.
Tchebichef, v , 33 , 47 , 50 , 54 , 70 , 1 00 , 1 86 ,
188,189, 414.
See Deviation, Polynomials.
Thome, 243, 422.
Titchmarsh, 37, 57, 301, 310, 311, 415.
Toeplitz matrix, 287, 399.
—, operator, 99.
Total variation, 12, 34, 35.
Transfinite diameter, 364, 369.
Tricomi, ix, 242, 243, 415, 422.
Trigonometric polynomials, 3,5 ff., 11.
— representation, 90 ff.
432
INDEX
Turan, vii, ix, 158,190, 388, 389,422.
See* Balazs-Turan, Egervary-Turan ,
Erdos-Turan, Griinwald-Turan,
Makai-Turan, Suranyi-Turan,
Szego-Turan.
Williamson, see Csordas-Williamson.
Wiman, 131, 416.
Winston, 131 , 351, 416.
Wintner, see Hartman-Wintner.
Wright, 203, 384, 416.
Uniqueness theorems, 248
Uspensky, 53 , 1 07 , 203 , 21 9 , 220 , 251 , 41 5 .
Young, W.H., 1 0,248,41 4 .
Van Veen, 132, 204, 415.
Vector, 9.
— space, 10.
Vietoris, 363, 422.
Vitali-Sansone, 415,423.
Vitali's theorem, 57.
Volterra equation, 211.
Wainger, see Askey-Wainger.
Walsh, 7,366,415.
See Sholat-Hille- Walsh.
Wangerin, 84, 415.
Watson, 18, 20 , 96 , 1 02 , 1 04 , 1 07 , 1 59 , 1 62 ,
192, 193, 202, 203, 222, 251 , 253, 363, 380,
384, 415.
See Whittaker-Watson.
Webster, 347,423.
Weierstrass, theorem of, 5 ff., 10,110.
Weight function, 9 , 37, 68, 1 59 , 1 60 , 1 6 2 ff.,
185, 188, 287, 294, 296, 297, 299, 309, 31 2 ,
314, 316, 347, 380, 385.
Weiss, G., see Coifman-G. Weiss.
Weyl, 110,251,310,415.
Whittaker-Watson, 1 4 ff., 58, 63, 65 , 66 , 71 ,
75, 84, 86,88,93,103, 248, 415.
Widom-Wilf, 243, 423.
Wigert, 33,102, 251, 413.
See Polynomials.
Wilf, see Widom-Wilf.
Zernike, 132, 416.
Zero-function, 9, 40, 45.
Zeros, distribution of, v , 310.
—, electrostatical interpretatio n of , 1 40 , 382.
- o f Airy 's function, 18,19, 377, 382.
analytic functions, 21.
Bessel functions, 1 2 6 ff., 1 40 , 1 92 , 1 93 ,
381.
Hermite polynomials, 11 7 ff., 123,
127 ff.,141 ff.,240, 353.
Jacobi polynomials , vi , 1 1 6 ff., 1 4 0 ff.,
144 ff., 192, 193, 23 7 ff., 25 0 ff., 379 , 381 .
kernel polynomials, 369, 377.
Laguerre polynomials , 1 1 6 ff., 1 2 2 ff.,
127 ff.,141 ff.,150 ff.,237 ff., 353, 381, 382 .
Legendre function s o f secon d kind ,
155 ff.
polynomials, 111 , 122,125, 353.
numerators of continue d fractions , 57 .
polynomials orthogonal on a curve, 369.
the unit circle, 292, 384.
solutions o f differentia l equations , se e
Sturm's theorem.
Tchebichef polynomials, 330,351.
ultraspherical polynomials , 1 1 9 , 1 2 1 ff.,
138,352,381.
See Orthogonal polynomials.
Zygmund, 248, 253, 254 , 274 , 276 , 279 , 281 ,
359, 416,423.

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